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Visualizing False Positives In Broad Screening
AlejoHausner writes "To find one terrorist in 3000 people, using a screen that works 90% of the time, you'll end up detaining 300 people, one of whom might be your target. A BBC article asks for an effective way to communicate this clearly. 'Screening for HIV with 99.9% accuracy? Switch it around. Think also about screening the millions of non-HIV people and being wrong about one person in every 1,000.' The problem is important in any area where a less-than-perfect screen is used to detect a rare event in a population. As a recent NYTimes story notes, widespread screening for cancers (except for maybe colon cancer) does more harm than good. How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?"
You mean.. sometimes these broads really are blond?
How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?
Hmm, teach them statistics?
"To find one terrorist in 3000 people, using a screen that works 90% of the time, you'll end up detaining 300 people, one of whom might be your target."
If you have a screen that works 90% of the time, and you detain 300 people, 270 will be terrorists.
There are Lies, Damned Lies and finally Statistics
The problem is important in any area where a less-than-perfect screen is used to detect a rare event in a population.
Such as "Who's a terrorist?"?
While it's true that there will be false positives, as well as false negatives, you don't convict someone, or have a lung removed, without further testing. When I was diagnosed for cancer, I was tested and re-tested to verify that there was, indeed, cancer. The same goes with screening for terrorists, or anything else. Did the article mention the rate for false negatives as well? After all, if you have a five pound tumor hanging off you face, and your doctor tells you there's nothing wrong, I'd definitely want a second opinion!
Just deny the insurance claims for the cancer screenings: "Sorry, you asked for a silly procedure. FAIL."
I want to delete my account but Slashdot doesn't allow it.
I'd say there's a one in a billion chance.
Not very clear wording. I would interpret that as 90% of those detained are terrorists. Which doesn't tell you anything about your false positive rate.
That's easy, just tell them that the screenings work about as well as speech recognition. It's 95% accurate and everyone knows how much it sucks.
How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?
You'll know when your people are ready for statistics. . . don't even bother trying until state-run lotteries go broke for lack of players.
I am not a crackpot.
Quick fixes for overweight people, or diet and exercise? Which do you think works?
What makes you think there's a quick fix for ignorance?
People who are not educated in statistics have only one solution which will work: Get educated in statistics.
That doesn't mean taking a maths degree. Any number of books or basic courses could help.
The problem is that most ignorant people don't want education.
And the really ignorant ones don't even believe that there's such a thing as being smart.
Just look at the number of people who play the lottery. Think any of them have the vaguest notion about statistics?
Think any of them WANT to learn?
Instead of giving a binary answer ("You are (not) a terrorist"), just communicate the probability ("There's one chance in three hundred that you are a terrorist. As this is higher than in the general population, we will investigate further to know for sure."). The same apply for medical exams. You can tell someone "The test tells us you have one chance in one hundred to have lupus. It's low, but higher than normal. We need to do more tests to know for sure".
http://www.amazon.com/Manga-Guide-Statistics-Shin-Takahashi/dp/1593271891
I hate math, always did. I was good at it but just could not stand it. As such I skipped out on about anything math related beyond algebra (college level). Didn't impede my programming ability at all.
Still there are times where I like to learn how stuff works and honestly this series of books, Manga Guide to ......, has given me a quick leg up on a few subjects I would never have gained from traditional text books.
* Winners compare their achievements to their goals, losers compare theirs to that of others.
It's obvious that the person telling us about statistics doesn't understand statistics.
On the subject of terrorism, why not simply arrest everybody, just to be sure...?
No sig today...
Though it may be the first time that people are trying to draw general population attention to it. I believe the first place I saw this sort of concept revealed was by Cory Doctorow. Though the below article isn't necessarily where I saw it, it recants the same message.
http://www.guardian.co.uk/technology/2008/may/20/rare.events
Charts. Big pretty colorful charts.
Chart one: Without this screening
# of this type of cancer found in last year, including false positives.
# of initial screenings determined to be a false positive in the last year, and number who were falsely diagnoses with this cancer after a false positive with a followup test.
# of deaths in the last year from the cancer.
# of deaths in last year from complications of treatment of people who were false positives.
# of man-years of productivity lost in past year lost to screening - typically 1-8 working hours per person.
# of man-years of productivity lost in past year lost to illness, and treatment, etc. from initial symptoms until person declared cancer-free or dies.
# of man-years of productivity lost in past year for treatment, side-effects, etc. for people who are false positives.
$ in direct costs for each of the above.
$ in lost productivity in the past year for each of the above.
Chart two: Same chart, but with numbers showing estimates if we had been using screening for the past 5 years
Chart three: Estimated annual increase or decrease in everyone's medical insurance and life insurance if we start using these tests per life saved.
People can decide for themselves if they are willing to trade lives saved for increased medical costs, assuming they go up. How much is a human life worth? $0.50? $500? $500,000? $500,000,000?
The moral of the charts:
Tests with a high false positive rate should be treated as initial screens only, not as a diagnosis. If a screen says "the general population has a 1 in 50,000 chance of having this cancer, but your chances are 50/50" or even "...your chances are 1 in 300" that's very useful information but not a diagnosis. A test that narrows things down to 50/50 might be worth spending real time and money on, a test that narrows things down to 1 in 300 may not be cost-effective by itself.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
Back during the TQM fad they'd make this point by giving everyone a clear plastic box with 10,000 little balls in it. There was a cribbage board like affair in it, with 1,000 holes, such that by inverting and shaking the box, then turning it upright, 1,000 of the balls would settle into the holes more or less at random, but still be visible through the clear box. The balls were color coded -- 10 red balls, 40 black ones, 50 blue ones, and the rest white. The odds of getting no red and no black are lower than 1%, contrary to most people's expectations.
This was used to drive home a point about the difficulty of "testing in quality" (quality tests suffer false negatives and if there are, say, 1000 such individual measurements on a piece of machinery it's nearly impossible to ship a machine without at least one thing wrong unless the tolerances are well controlled at the point of manufacture). The same idea works any time you want to illustrate the effects of low-incidence events on a large population.
I've always wondered how much injustice is perpetrated by drug screening on large populations, since false positives do occur and statistically must occur twice in a row at least some of the time, which is the threshold considered conclusive proof of abuse by most employers and the courts.
Not sure about other places so I'll speak for the US here. If you could have detected something (terrorist, cancer, etc), but didn't, you'll get taken to court for it. If you run tests you're covered no matter what the results. If you didn't, you're negligent and will spend the rest of your life selling paperclips out of a cardboard box. As an off-topic aside, these ridiculous lawsuits are why healthcare in the US is as bad as it is and no administration is going to admit that or fix it because putting lawyers out of work is the last thing any politician wants to do.
mmmm...forbidden donut
It can't. People are stupid; Americans doubly so. We live in a country where half the people believe that a talking snake catered the wedding between a naked man and a naked woman made out of his rib, which enraged the upstairs landlord so much he evicted them. They also believe some guy sitting on a sky chair gives a shit how a high school football game ends. They believe that the Honolulu paper printed a fake birth announcement in 1961 so that 48 years later a socialist elected to take away their guns. And those that don't? They think that Kennedy was shot by aliens hired by the Illuminati and that a president blew up some office buildings in 2001 to steal the gold in the basement. Idiots all.
You want these idiots to understand how statistics works? Then you need to write a Java program that demonstrates it so simply that a retarded chihuahua could understand it. Oh, and it has to be colored red, white, and blue, cause this is 'Merica!. And it has to have tits in it somewhere.
"I guess the moral of the story is, don't paint your airship with rocket fuel." -- Addison Bain
Better yet, how can efficiency be explained to wannabe nerds? If a test is very fast, non-intrusive, and cheap with a 90% accuracy, it is a great test. Those 10% may be sent to a further test that is longer, more intrusive, and more expensive with a 99.999999999% accuracy. This applies throughout testing of all kinds. There is no reason screening for terrorists should be a magical area of testing separate from the rules that govern all other areas of testing.
The previous comment is purposely vague and generalized, but all of the facts are completely true.
Here you go: http://www.bmj.com/cgi/content/short/327/7417/716?fmr It is titled: "Understanding sensitivity and specificity with the right side of the brain "
Exactly written for the purpose. PDF should be available freely.
The article itself started out by oversimplifying the test. It would be an astounding coincidence if the test had both a 10% false-positive and a 10% false-negative rate. In fact, any normal test has a very different false-positive and false-negative rate. People who describe the test should mention both, not this meaningless "90% accurate" number.
The BBC article, while claiming to want to reduce confusion, actually perpetuates the problem by using the meaningless "90%" number instead of the specific positive and negative failure rates. If every article describing tests would quote both failure rates, that would go a long way to getting people to understanding the situation.
I hate it when I make a joke and I get modded "+5 insightful". Mod the stupid comments "funny", not "insightful", pleas
Yeah, and seatbelts cost more lives than they save.
It may cost more money to institute screening programs and chase false positives, but cancer survivability numbers have been increasing steadily for the last forty years BECAUSE of early detection, not in spite of it. In fact, I remember a statistic from a few years ago that showed more people were surviving all cancers in straight numbers, not just per capita. That's some increase!
I'll take the possible side-effects of false positives over the known repercussions of no positives any day.
I'd give the unschooled massed a table like this:
For every 1000 people you scan, you will have:
20 innocent people who are identified as a terrorist.
1 terrorist who is identified as a terrorist.
1 terrorist who is not identified at all.
Simple, can be read quickly, very easy to understand.
Of course, the unschooled masses will still do stupid things like, "Hmm, the first 19 people were innocent. This guy must be a terrorist."
"Works 90% of the time" here means that it will correctly identify a person as terrorist or not-terrorist in 90% of tests.
On a sample of 3000 with accuracy 90%, you will end up with 300 results guaranteed to be wrong or ambiguous, which may or may NOT mean the subject is a terrorist. To be safe, obviously you have to detain these "ambiguous" subjects.
Considering that we know the number of terrorists is incredibly small (from a UK perspective, I'd say something like 100 in 70 millions, or 1 in 700.000, probably even less), we can deduce that these tools are guaranteed to victimize thousands of innocents (at least 69.999) for each "terrorist" ever caught.
-- Let's go Viridian.
To find one terrorist in 3000 people, using a screen that works 90% of the time, you'll end up detaining 300 people, one of whom might be your target. A BBC article asks for an effective way to communicate this clearly.
Innocent until proven guilty. As in, you're not a terrorist until you do it. I suppose there's some lame, "with intent to blow himself up" garbage in law, but that's irrelevant IMHO.
"but money is the God of Algiers & Mahomet their prophet." - Rich. O'Bryen June 8th 1786
Screening tests should not be called "90% accurate."
They should be specific: "This tool's declaration that you are not a terrorist is 99.99999%+ accurate, it's declaration that you are a terrorist is 0.3% accurate."
Compare this to the "control test" that declares everyone a non-terrorist: its 99.9%+ accurate in identifying non-terrorists and 0% accurate in identifying terrorists.
The other "control test" which identifies everyone as a terrorist is 0% accurate at identifying non-terrorist and 0.00...001% accurate in identifying a terrorist.
What we need is a green/yellow/red system:
Green means we are much more sure you are clean than if we did nothing AND we are at least 99.9%* sure you are clean.
Red means we are are much more sure you are not clean than if we did nothing and we are 99.9%* sure you are a terrorist.
Yellow means something in between, with a specific level of confidence assigned to everyone in the yellow group. Those with higher confidence of being clear would undergo less additional scrutiny than those who have a higher confidence of being a terrorist.
*The numbers should be tweaked to suit the needs of the test. For terrorism screenings, where you don't want to outright ban anyone from flying who isn't a danger, the red level should be high. For cancer, it might need to be lower, or higher, depending on the specific patient needs. The yellow discretionary band should be very wide for most applications.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
former President-VICE Richard B. Cheney.
I hope this helps the legal effort for prosecution at The Hague.
Yours In Justice,
Kilgore Trout
Combined with AUC (Area under the curve plots), this is the point of ROC curves, and confusion matricies: they are ways of comparing True to False positives
http://en.wikipedia.org/wiki/Receiver_operating_characteristic
This is covered in any decent stats book since the 60s'...
Anyone can write software to look for a turban
This sort of racist bollocks is what has been getting people attacked in the US for wearing them, despite them being an optional part of the Muslim faith so most turban wearers are from entirely different religions which actually require them.
http://en.wikipedia.org/wiki/Turban
Please educate yourself before posting such drivel.
I dont read
...you cant! A populace that (for the most part) relies on intuition for day to day operations dismisses counter-intuitive facts with ignorance ("That does not make sense") or arrogance ("It will save more lives that it hurts, so quit twisting the numbers around!"). People all too often forsake science for "cereal box science."
1331461 is only semiprime *sigh* Alas - I am just short of 1337.
The given version of "terrorist" is arbitrary and thus subject to change over time - from people who hijack planes with guns and explosives, to apparently nowadays, Iceland, however I think that if you're starting with a number of 1 in 3000 you are so far from reality anyway that what you really want to do is harass innocent people.
Let's look at ALL the hijackings from 1970 to 2000, a total of 924 hijackings. I couldn't find more recent figures quickly, but let's assume that hijackings have continued at a rate of around 30 per year (the average from 1970-2000), that would add another 30 * 9 = 270 hijackings, for a total of 1194 ok I will be generous 1200 hijackings.
Now let's assume (and this is a BIG assumption - I am again going to be very generous) that TEN people, (the terrorists), board the plane for EACH hijacking event. So now we have 12,000 terrorists.
Now let's just look at the passenger data for the LAST YEAR ALONE for the top 5 airlines. They carried last year 420 million people. LAST YEAR. Now assuming that since 1970 till today there have been a total of 12000 "terrorists" (a VERY generous number), when you divide 420 million by that, you would be looking at 1:35,000 people being a "potential terrorist". However do remember that I am only including passenger data for ONE SINGLE YEAR. Assuming again a 90% accuracy, you are still wrongly intimidating well over 3500 people.
If I was to go through year by year and gouge up the billions of people that have been transported by air, the actual chances of the person being screened actually being a terrorist drops to almost zero.
I will not argue against the value of security as a deterrent. However I think that airport security employees should be well aware that they are, more likely than not, harassing innocent people. Therefore all the excessive bullying, posturing, abuse, privacy and rights violations are completely unnecessary in this context. Airline terrorism is NOT a real threat, be it ever so dramatic on the few times when it does happen. Use technology to screen for the obvious, and lock the god damned cockpit door with a solid lock, for the not so obvious.
Seven puppies were harmed during the making of this post.
"The problem is important in any area where a less-than-perfect screen is used to detect a rare event in a population"
Unfortunately, there is no such thing as a perfect screening test for anything in medicine. Some are better than others, but none are perfect. This is a very difficult concept for most people, unfortunately, and for many insurance companies.
It is not such an issue for the better screening tests such as colonoscopy but it is very difficult for things like PSA where there is a large body of evidence it can do more harm than good on average if used routinely even within the recommended ages. For a patient, you're lucky if you can have a meaningful discussion in 5-10 minutes which is an awful large chunk of an office visit that usually has >4 talking points.
It is a problem for doctors and insurance companies because some well intended person with the insurance company will decide to measure the quality of its doctors (which I support in theory) by measuring, for instance, the percentage of age and gender appropriate patients under the care of a given physician that have their PSA checked annually. The problem is, there is absolutely no concensus in medicine that it should be checked regularly as a screening test. I'm not sure I want mine tested when the time comes around unless my family history changes between now and then. So to measure a physician by this marker or other screening tests is fraught with problems, since many patients might opt out for very good reasons. Also, I'm not going to recommend any test because an insurance company wants me to, only if it is right for any given patient.
Bottom line is there are no perfect tests and testing is not always the right thing to do. Most people do not understand that because it is a hard concept to grasp.
"""To find one terrorist in 3000 people, using a screening that works 90% of the time, you'll end up detaining 300 people, one of whom might be your target.""" ,dictates that 10% happening when the terrorrist passes.
Does that mean that the terrorist has a 1 in 10 chance of not getting flagged? If it works 90% of the time that means it doesn't work 10% of the time. And we all know that, 'enter some dudes name which I very coinsidently forgot right when I needed it!!!'s law
That means you detain 300 people, and none of them is a terrorrist. Added to the fact that Americans are paranoid as shit, and there's not even a terrorist in 3 million people, you just found a way to create many jobs in 'Homeland Security'. And ofcourse a new way to piss many people off.
At least I know that my taxeuros are being used to fix space toilets.
P.S.: the guys name is Murphy, now I remember.
I'm so glad you posted this. First I was neutral about turbans. Then I read that guy's post and I immediately concluded that turbans are worn only by terrorists. Luckily, I then read your post, and was thankfully corrected. Good save.
Maybe a new slogan:
Precede with an explanatory: "Accuracy ratings for screenings are not based on how many of your target you find, but on how many false positives you avoid."
then hit em with: "This screen is 90% accurate! 90% of the people who pass through here will not even be considered for waterboarding and sexual humiliation!"
or maybe an illustration:
"You live in a town of 3000 houses, 911 receives a call reporting a fire at your house number. There are 30 houses (1% of the total) with the same number. The fire company breaks down the doors and smashes the windows of all 30 in random order. All valuables are destroyed by water damage, and just by coincidence your house burns to the ground.Their screening method was 99% accurate, and in fact their method of finding your home was 100% accurate."
or a simple ascii of 2999 zeroes with a 1 hidden among them..... I actually made this and the 90% and 99% and 99.9% versions; but slashdot won't let me post that many repeating characters.
Draw a picture. People's visual intelligence is much higher than their literary/verbal intelligence. Descriptions in words are difficult to understand when the meaning of the words being used is not clear, uses domain specific jargon (such as 90% accuracy in relation to statistics) and especially when it requires that the recipient of knowledge perform a mental calculation or solve a mental equation.
An effective picture would be one of a thousand people (stick figures or silhouettes will do) with 10 positioned in front. A caption over the 900 in the big group would say "Tested Negative (These people are NOT Terrorists), the caption under the 10 in front would say "Tested Positive (These people may or may not be Terrorists - We don't know)".
Then ask people how they would feel if they were in the group of 10 and were going to be shipped off to a military holding cell to await further investigations.
A fool throws a stone into a well and a thousand sages can not remove it.
He's dark-skinned and his name is Mohammad.
(Wait, are we talking about airport screening or HIV testing?)
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
I think they prefer the term "mammogram".
Say there is a cancer that affects 1 in a million people. Say its typical symptoms could mean cancer or one of several other diseases. Say the currently available test is highly accurate, with few false positives and negatives, but it is highly invasive, highly costly, damages healthy tissue, and has a high risk of complications. Say there is another cause of the symptoms which is non-progressive and non-fatal but which there is no test for. The standard treatment is to test for everything you can test for, then if all other causes come up clear, watch to see if things get worse, and if it gets worse, do the cancer test. Statistically, about 25% of the people with these symptoms have the cancer. If you start treatment now, the 5-year survival rate is 90%, if treatment is delayed until symptoms get worse, it's 80%.
Now say someone comes along with a cheap, highly accurate blood test that can say with a 50/50 chance that you carry the gene for this type of cancer. Let's say those without the gene almost never get this cancer, and those that do have a 10% lifetime risk. Hurray, it's a no-brainer. You take the test, if you do not have the gene, you breath a sigh of relief. If you do, you decide if you want to do the invasive test, knowing there's a significant chance it will be positive, but you can begin treatment now with confidence you are doing the right thing.
But what if the new test is not cheap, easy, and highly reliable? What then? That is the crux of this issue.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
I think they totally forget that there is ALSO a 10% possibility that you _don't_ detect the terrorist...
Watch this TED : http://www.ted.com/talks/peter_donnelly_shows_how_stats_fool_juries.html
Privacy is terrorism.
Just anal-probe every American at football games and airports, then tell a random 9.99% of them they've got AIDS from the procedure (though they don't).
We'll develop an intuitive sense of empathy for falsely accused Muslims, other turban-wearers and lots of other people "with nothing to hide".
--
make install -not war
Where do you start? What do you start with? We're foveated animals with brains that seem to have much to do with 'where' and 'what', or, 'object' and 'place'. That's a ridiculously broad epistemology to try whittling down to a point and click understanding of statistics. Do people not understand statistics, or, do they not understand sampling and false positives; or, do they just not understand being detained, strip searched and subjected to a body cavity search, no matter the underlying science. Statistics is fairly intuitive and, as such, is amenable to the fundamental maths' concept of inspection. Inspection as I understand it is alot like Yogi Berra saying you can observe alot just by watching. If you buy fresh, bagged vegetables you pretty much understand sampling the visible attributes and inferring the nonvisible ones, and, anyone whose done so has gotten home and found at least one bad apple (fruit, vegetable, whatever). Buy the same brand of bagged vegetables and discover what you consider to be a disproportionate number of unseen, unwanted attributes and you'll switch brands. The hidden problem maybe that people think they're being flagged by a machine, by an AI, and, maybe they are, but they'll most likely be apprehended by a person and it may be that the point and method of apprehension is where a quick explanation needs to be inserted, like being Mirandized, only explaining "the computer" has selected them based upon yada yada... Maths, to most people is anathema (anathemathtic). I recently undertook a review of maths via, Jason Gibson, NASA rocket scientist, and his MathTutorDVD stuff plus a bunch of other stuff. How do you explain maths is mostly straight forward to people who have what amounts to a mental block against abstracted symbol manipulation? The human limbic system incorporates our reward system and that system is pretty wet stuff, not too interested in abstract logic. Recently intelligence has been liked to networked neurons in the association pathways. This may indicate that most healthy, normal people intuitively and rightly avoid intelligent explanations and intelligent people in favour of wet, warm, limbic, puppy love. I hold we're formulating a new mythology wherein the old Dionysian way of ecstasy most famously iconographed by the Dionysian sacred, disease of epilepsy is being replaced by the new sacred of Asperger's Syndrome exhibited by dry, maths types. It's just a matter of a few thousands of years of evolution.
ideopath @ play
Lol.
Who said something about muslims?
You've been trapped by the same fallacy as that other guy. You're scolding him for using the turban wearing terrorist stereotype, but using the equally narrow minded muslim terrorist stereotype.
Plus, completle ignoring that it was an obvious joke about (drumroll) stereotypes!
bickerdyke
"How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?"
With a baseball bat?
If the cancer screening is less than a couple month's pay and the illness runs in the family, many people will pay out of pocket.
If it's less than a day's pay a lot more people will pay out of pocket even if they are at nearly zero risk to start with.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
Switch around what the percentage means: instead of 90% meaning there is 90% chance to successfully ascertain whatever you're screening for, make 90% stand for the analogy of LD50 (Lethal dose for 50%+ of the population). So the screening method would be SE50 (screening effective 50%) if the number of positive cases correctly detected are 50%+ of all positive cases detected.
It takes a man to suffer ignorance and smile
Be yourself no matter what they say
Yet you failed to learn from that post that making insulting jokes about how anyone wearing a turban in the US can be beaten senseless "because they're a Muslim terrorist" is unacceptable in public (and probably in private, depending on the quality of the company you keep).
--
make install -not war
Barring twins and clones, when done properly and with a very good sample, a complete DNA test has a lot higher than 99.99% correct-positive rate and except in cases where a person has multiple DNAs due to an organ or tissue transplant or congenital factors, a 100% correct-negative rate.
When done improperly or with a corrupted or planted sample, all bets are off.
Unfortunately, the DNA tests done in court are far from complete. They are still better than 99.99% when done under ideal circumstances, but are less accurate than a complete test.
Fingerprints done under ideal circumstances are quite good at pinpointing a suspect, especially if the list of people who had access to the crime scene is in the thousands or lower a "planted" fingerprint is ruled out. If your fingerprint is found in the blood spatter of last night's murder, you better be able to explain why you were there after the blood was.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
A family with two children is chosen at random from a large population.
If I tell you only that they have at least one daughter, what is the probability that both children are girls?
Most people can get that one (it's 1/3), but fail miserably on this question:
If I tell you only that they have at least one child named Mary, what is the probability that both children are girls?
Assume the obvious: the boy/girl ratio is 50-50 and only girls are named Mary.
Most people insist that this is the same question with the same answer, but no, it's not, and the answer is actually 50%.
If you don't get this puzzle, you don't understand conditional probability.
"Skill shows through where genius wears thin." -Wittgenstein || Religion: uniting aviation and architecture.
You'd have a really good point if there weren't actually bigoted assholes and/or ignorant people in the world who agree with the great-grand-parent. Earlier in my life I may have been one of them.
I remember painting Muslims with a very broad and unfair brush. People would tell me that all Muslims aren't bad and most want the same thing I do, peace and prosperity. Why don't they speak out against the bigoted extremist representatives then? I would ask.
I didn't have the slightest understanding of the culture and environment those types of ideas breed in and probably still don't. However, I can come out of my own bubble enough to ask myself the question - What motivation would I have to speak out against wrongs being done against a culture who shows repeated disrespect and ignorance for my own?
I'm not suggesting we adopt sharia law and that all North American women start wearing burqa as a sign of respect. There is a very thick line between embracing and adopting a culture and respecting it.
So China's population is 1,331,950,000 according to Wikipedia today.
Also from Wikipedia, the world's population is estimated to be 6,773,000,000.
This means that about 19%, or nearly one in five people in the world, is Chinese.
There are six people in my family, so clearly at least one of us is Chinese.
Turbans are worn by Sikhs. This is a completely different religion to Islam which is alleged to harbour these terrorists.
Well, I wouldn't expect most people to intuitively grasp Bayesian statistics without some formal introduction to the subject. Nevertheless, it must be remembered that as part of a decision algorithm for detecting terrorists/cancer/whatever, a not-so-accurate test can be useful as a first step. Specifically, if said test is minimally bothersome, cheap and permits us to apply a more costly and accurate strategy to a limited number of individuals at a further stage. For example, a metal detector stops about 30-50% of passengers, very few of which are terrorists. Subsequently, some will get strip searched, some will be detained and some will enjoy a free anal probe. Although failing a metal detector is not catastrophic, failing all subsequent (progressively "invasive") examinations is highly suspect (accumulated evidence).
So, a 90% specific (false positive) test is not worthless if it is sufficiently sensitive: it protects 90% of the population from further tests and saves money.
P.
It's good to inform people who don't understand statistics. On the flipside, here are two points for people unfamiliar with security:
1. A broad screening for "terrorists" is not made with the expectation that every person flagged is a terrorist. Rather, it identifies behaviors that make a person worth giving a second look. If properly conducted, the flagged person is not treated or considered a threat during the second or even the third look. The 300 people you mentioned would almost certainly be treated politely and sent on their way (I myself have received a second or third look several times. No problem.)
2. Perhaps the most important purpose of a broad security screening is to discourage criminals from using that avenue in the first place. If I have several dozen potential means of attack, for example, the ones that involve getting a weapon onto an airplane are going to be near the bottom of the list. Not because I can't do it, but because, why bother?
Evil is the money of root.
This sort of racist bollocks is what has been getting people attacked in the US for wearing them
What a load of crap, back in 2006 NBC dateline had a bunch of muslims go to a Nascar race and see if they were harassed, guess what they were NOT bothered at all. This sort of idiotic bollocks is what perpetuates the myth that the US is full of racists.
Knowledge = Power
P= W/t
t=Money
Money = Work/Knowledge so the less you know the more you make
I'm not suggesting we adopt sharia law and that all North American women start wearing burqa as a sign of respect
Of course not. They should wear turbans.
that's because it's a well known fact* that ter'rists hate Nascar. It's also a well known fact** that ter'rists won't pay to get in to blow someone up. thus we can deduce with 100%*** accuracy that these guys could only be ter'rists in the breif period from when they left their car in the parking lot and walked to the gate. Furthermore, that they parked their car means they likely were not ter'rists because all ter'rists**** blow their cars up.
-nB
* And I'll put it on the internet if you ask me to quote it, and since it's on the net it must be true+
** again, if you call me out on this I'll just post it somewhere to quote it.
*** selective sample set (lies, damned lies, and statistics, right?)
**** just seemed good
+ I saw it on the internet somewhere, seriously.
-nB
whois gawk date unzip strip find touch finger mount join nice man top fsck grep eject more yes exit umount sleep dump
One way to do it is to frame it not as accuracy, but simply as candidate reduction. It reduces the size of the haystack by 90%. We're still looking for a needle in a haystack, but now the haystack is one tenth the size it originally was.
guess what they were NOT bothered at all
Yep, no hassle at all - in fact most people didn't even get closer than the blast radius from a decent sized stick of dynamite!
which is totally what she said
Why don't they speak out against the bigoted extremist representatives then?
The same reason huge swathes of the U.S. population didn't speak out against the bigotted extremist representing them between 2000 and 2008.
They're dressing like Sikhs to throw us off the scent! But us americans are smarter than that, we see the turbans and know!
Fooled me can't get fooled again!
There you go... no charge.
She was like chocolate when she drank... semi-sweet at first and then increasingly bitter.
To find one terrorist in 3000 people, using a screen that works 90% of the time, you'll end up detaining 300 people, one of whom might be your target.
Anyone can write software to look for a turban
Let me see if I got this straight. If I write a piece of code that detects turbans, I'll pick 300 out of every 3000 people, of which 1 might be a terrorist? I'm confused now.
The basic problem is not with the screening process, but simply knowing how many "actuals" there are out there to be caught - or deterred. To put it another way: how do you know when to stop? All these security measures assume that there are still terrorists with malevolent intentions trying to get on airplanes. As a concerned passenger, I am all for stopping them provided they still exist. However if I am being put through the whole circus of intrusive and inconvenient security restrictions when the threat disappeared long ago, I would like them to stop. Merely having some official who's job, budget and political power rests on prolonging and talking-up the level of threat, saying "Trust me, I'm in the government" is not sufficient and not credible. We need to know actual facts: who was caught, what were they trying to do, what would have actually happened if they had succeeded.
I have a sneaking suspicion that even the people in security don't know the answer to this. While they may read reports to say "X number of people were detained last month" and assume from that, that the threat is real or increasing or that they're "winning", there are no hard, auditable facts available to back up these hypotheses. If that is the case, then the measures they have put in place are probably not even the best response to the level of threat. It seems that they are basing the level of security needed (which is also variable from place to place: flights entering a country are subject to vastly different security checks from flights leaving a country) on guesswork and the probably incorrect assumption that it's working since there haven't been any more September-11 type incidents. This attitude just plays into the hands of the baddies, as they continue to strike fear into people with little or no effort on their part - while still being able to come up with new and novel strategies as the established security measures are still fighting last year's war.
politicians are like babies' nappies: they should both be changed regularly and for the same reasons
Of course we should not beat up people wearing turbans of all colors. Remember muslims only wear white or black turbans. If they are of any other color, they're not muslims, probably sikhs.
What a load of crap, back in 2006 NBC dateline had a bunch of muslims go to a Nascar race and see if they were harassed, guess what they were NOT bothered at all.
So you think the following disproves the existence or danger of grizzly bears?
What a load of crap, back in 2006 I took my kids on a camping trip to see if they would be eaten by grizzly bears, guess what they were NOT bothered at all.
To quote Bugs Bunny, "You, sir, are a maroon!"
There's a difference between a test that identifies 99% of all terrorists, and a test that identifies people as terrorists, 99% of whom are really terrorists.
Perhaps a good way to teach statistics would be to start by teaching it to BBC reporters.
It's a non-problem. There are math teachers who know how to explain this very clearly. And there's a lot of research on how to communicate the results of medical tests to patients.
There's a brilliant article that discusses this problem here: http://yudkowsky.net/rational/bayes. About a quarter of the way in there's a look at a few different ways of expressing the quantities. It seems frequencies are good (3 in every 1000 innocent people will be IDed as a terroist or 299 in every 300 people identified as a terrorist are innocent). People intuitively focus on the expected outcome - postive test result == terrorist and negative == not a terrorist. Maybe the way to make it clear is to tell them the non-intuitive statistics (299 in 300 that appear guilty are innocent while 1 in 30,000 that appear innocent are guilty). The issue is that if you tell someone "Q given P" (positive-result given is-terrorist) they always fall into the trap of thinking "P given Q" (is-terrorist given positive-result). Saying the test is 90% accurate is saying "Q given P 90% of the time". No one understands prior probability yet figures like this always ignore it.
http://en.wikipedia.org/wiki/Bayes_theorum
See the section on drug testing.
Praying for the end of your wide-awake nightmare.
How about evacuting a city several times because of a hurricane but it misses?
How about spending a lot to avoid global climate change that never happens or does not turn out to be as bad as predicted by our current understanding?
The only way I can think to explain this to the "general public" is to make the analogy with insurance. A cost you incur but probably never get any benefit from.
I thought the text you offered just then was pretty good.
The 9/11 and 7/7 attacks etc were instigated by self proclaimed Muslims. There's no "alleged" about it, it's just a fact. Note that I don't believe that all Muslims are terrorists, or that all terrorists are Muslim, that's just stupid. But the most widely publicised and recent terrorist attacks have been strongly linked with Islam. That doesn't mean that Islam "harbours" terrorists, a lot of Muslims don't agree with these violent attacks. But it's pretty safe to say that the majority of attempted terrorist attacks within the next few years are likely to be instigated by Muslim fanatic groups who are pissed off about the whole invasion situation in Aghanistan/Iraq/Iran. The 7/7 attacks here in the UK were "carried out by 4 British Muslim men who were motivated by Britain's involvement in the Iraq War" according to Wikipedia.
which is totally what she said
The simplest way to get a message across to the "masses" is simply to have a celebrity deliver it. No explanations, no demonstrations. Simply a script that says: "you know me, I'm that nice, trustworthy person from <name of popular programme> so you know when I speak, I'm telling the truth ..."
People tend to trust individuals they know, they "know" the characters on TV - even though they are actors and probably nothing like that in real-life. It doesn't matter, just think about all the causes that get a celeb on board and then effect political change, even though it's a tiny (but vocal) minority of the population involved and therefore about as non-democratic as it's possible to get.
politicians are like babies' nappies: they should both be changed regularly and for the same reasons
I mean there are so few, incalculably few, terrorists that the chances of getting two, unconnected ones on the same flight are vanishingly small. The chances are probably less than for the whole plane to be abducted by aliens. So if you supply one, the odds of a "real" one drop to nothing, in practical terms. Problem solved!
p.s. for the comically inept, this is not meant as a serious suggestion. I suggest you start breathing again before you turn blue, or purple, or republican.
politicians are like babies' nappies: they should both be changed regularly and for the same reasons
There is a branch of statistics called Decision Theory which uses expected values which have a subjective weight. Thus the value of a dollar, say, may not be the same at two points in the equation.
Parity: What to do when the weekend comes.
Compare the test result accuracy to the accuracy of typical spam filters.
People get spam, and understand the problem with false positives and false negatives.
If you say "if your spam filter was as accurate as this test, you'd have an XX% chance to lose mail from your bank or your spouse, and a YY% chance to get Z pieces of spam anyway", I think that would make the numbers considerably more meaningful for a lot more people.
How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?
Screening for HIV with 99.9% accuracy? Switch it around. Think also about screening the millions of non-HIV people and being wrong about one person in every 1,000.
I wonder if it's possible you mis-stated the problem? If you tell me they have two children and that at least one is a daughter, the tested condition ("both children are girls") comes down to the other child also being a girl, with a probability of 50%. If the other child is a boy, then the condition is not true, and if the other child is a girl, the condition is true. The condition will be true in 50% of the cases. Am I missing something?
Put another way, "In all families with exactly two children, both children will be girls in 50% of families who have at least one girl."
I'm assuming, as you stated, that any given child has an equal chance of being a boy or a girl.
Evil is the money of root.
So even if the security people who make these bald, and unsupportable claims *did* have a large enough number of caught terrorists in, say, 2002 - the chances are that the number has dropped significantly now. Of course, once you start to reduce the level of security, the number of "incidents" will probably rise - although as said, not in the way that the security people are prepared for. All this tells yo is that the statistics are meaningless - however you view them. All they can do apply a level of security / restrictions that people, en-masse are willing to tolerate and hope that's enough.
politicians are like babies' nappies: they should both be changed regularly and for the same reasons
A nurse recently told me a test was never wrong, it was 99% accurate. I asked how many people she had used the test on this week, she said about 50 a day. Without any knowledge of the population % of positives, and making the gross assumption that it was 99% false positive and 99% false negative, that would lead one to believe that she seeing incorrect results about 2-3 times a week. This had simply never occurred to her. Never mind the population statistic, nor the possible difference between false positive and false negative, but she understood 99% accurate to be - "never wrong". It never even occurred to her that if she was testing hundreds of people a week that some results would be just plain wrong. I didn't ever bother trying to explain the effect of population statistics.
http://davesboat.blogspot.com/
*WHOOSH!!!*
Bruce has blogged several times over the years, about the problem of false positives:
Here are a couple of them:
http://www.schneier.com/blog/archives/2005/01/terrorism_false.html
http://www.schneier.com/blog/archives/2005/03/nuclear_terrori.html
http://www.schneier.com/blog/archives/2009/04/dna_false_posit.html
Please explain this assertion. I'm having problems getting past "since only girls are named Mary, 'at least one child named Mary'=='at least one daughter'". I understand that the converse is not true.
Yet you failed to learn from that post that making insulting jokes about how anyone wearing a turban in the US can be beaten senseless "because they're a Muslim terrorist" is unacceptable in public
Unacceptable to whom? And why?
As I read the joke, it is making fun of ignorant American rednecks. While I guess some people might find painting such people with such a broad brush insulting or offensive, I don't see it, myself.
Apparently you a) have a different interpretation of the joke and b) feel that your interpretation justifies declaring the joke "unacceptable in public." I don't get your logic, and I certainly don't appreciate your arbitrary and unjustified declaration regarding what is or is not acceptable behaviour "in public".
This is particularly true since /. has a significant world-wide readership--if you clowns can't control your bigots that's your problem, not justification for declaring, in typically American imperialist fashion, what is and is not acceptable here in this international, albeit US-dominated, forum.
Ok, the problem with this comment is that it is now a) exactly what I feel and b) -1 flamebait. Oh well.
Blasphemy is a human right. Blasphemophobia kills.
After all, "not all KKK members are terrorists" is certainly a truth.
Please tell me, should one be tolerant against people who have an intolerant ideology, as indicated by said ideology prohibiting inter-group relations and promoting attacks against outsiders, like the KKK does ... or islam does ?
Anyone who believes
O you who believe, do not take Jews and Christians as friends; these are friends of one another alone. Those among you who ally themselves with these belong with them. GOD does not guide the transgressors.
(quran 5:51) is not a tolerant individual. Any muslim who treats any non-muslim in a friendly manner commits treason. In a number of nations this will get you the death penalty, especially for women, due to islam specifying it.
Since this is part of the most essential part of islam, anyone who does NOT believe this is not a muslim.
Ergo one is either religiously intolerant Xor one is not a muslim.
Being a tolerant muslim is like being an innocent murderer, a vegan game barbecue : a contradiction by itself.
Should you be tolerant against people who believe this is the way to live ?
Of course, since racism and religious intolerance is a core part of islam itself, that is the exact same question as, should you tolerate muslim ideology ?
So, please tell me, which of these is true :
1) we do not allow for religious intolerance, and therefore we do not allow muslims and/or islam
2) we allow for intolerance, in a non-racist way, and we show the same respect to KKK members as to muslims
3) we allow intolerance in a racist way : some groups get to be racist, others get punished
Obviously 3, your choice, is the worst of all in the effect it'll have on racism. It both makes you yourself a racist and encourages racism in others.
Needless to say, we have apparently chosen the dumbest of these options, number 3. Of course, before 1941 there was no politician in public office who dared call Hitler anything but the "champion of the poor".
The ideology of islam is as much an enemy of human rights, of equality and of tolerance as nazism. Please treat it as such.
To those who don't see the 1/3, consider 2-child families with children listed oldest first:
1/4 have Boy, Boy
1/4 have Boy, Girl
1/4 have Girl, Boy
1/4 have Girl, Girl
You know there is at least one girl, leaving 3 options. One of those has two girls, so you have 1/3.
As for the Mary thing, the best I understand is if you say the FIRST one is named Mary, since only two options have a girl first, making it 1/2 for having two girls. If I am missing something please correct me, but I honestly cannot see how you get 1/2 without specifying which child is Mary.
My webcomic
That was the sound of a joke going over your turban.
Facebook is the new AOL
Um, yeah, forget what I said about 50:50 in the first case. But I too am having trouble seeing the difference between the first case and the second case.
Evil is the money of root.
Yes, the software should analyze skin color and accent.
Your second example is only true if you add an additional assumption: that a family never has 2 children with the same name, in which case your question is better expressed as "If I tell you only that they have exactly one child named Mary, what is the probability that both children are girls?". In this case the probability is 50% as you point out.
However, if a family can have 2 children with the same name the probability will depend on the proportion of Marys amongst girls. For example if all girls are called Mary, your second question *is* the same as the first one and the probability is 1/3. If Mary is a rare name (as in the real world), the probability will be slightly less than 50%.
It's a matter of precision and recall, a very elementary and important concept of statistics.
Basically, anyone who designs or uses any kind of test designed to find the proverbial needle-in-a-haystack ought to learn some basic fscking statistics and how they apply to this situation. And yes, that includes police officers and screening personnel in the field, in my opinion.
My bicyles
This seems like a good argument to get at least an overview of statistics into the standard high school curriculum in America.
Okay, if you don't get the problem (and yes it is correctly stated), there are three things you can do to improve your understanding.
1) Write a simple computer program in your language of choice to generate a billion random two-children families and count the ones that meet the conditions.
2) Draw a Venn diagram.
3) Read up on Bayes' theorem.
"Skill shows through where genius wears thin." -Wittgenstein || Religion: uniting aviation and architecture.
if you read about how medical tests are done, there is ahuge literature on this, and it makes the point of the article, that teaching this stuff, is not easy
see the wiki article on negative predictive value - its all there, just not clear
This is astroturfing. It is no accident that the South Capitol Metro station (beneath the U.S. House offices) in Washington DC is currently wallpapered with PRO-Cancer-Screening posters.
Considering a $100 test, a $1k treatment and a 5% actual frequency in the population, per $1K you're spending about $50k on +/+, $100K on -/- and an inconsequential amount on BOTH false positives AND false negatives.
HOWEVER, outside of that you have untested treatment, often prophylactic. This happens a lot with my insurance carrier. Diagnostic testing denied coverage almost 100%, treatment denied without diagnostic testing or overt presentation of symptoms. Now, given the same frequency, uncovered, untested conditions would account for about the same $50K in expenditure for +/+ and if just 10% opted to go ahead with treatment prior to becoming symptomatic, it would generate the same $100K in prophylactic treatment -- out of pocket. The false-negatives in that case become VERY cheap indeed, being, well, dead.
Is it any wonder they are discouraging testing?
I think that's called the "prosecutor's fallacy." If there's a 1/10,000 chance of a child dying of cot death, and a woman has two children die of cot death, the prosecutor tells the jury that the chances could only be 1/10,000 * 1/10,000 = 1/100 million that both deaths were a cot death, so she must have murdered them.
This only works if the deaths are statistically independent, which they're not. The parents could have a genetic defect which cause 2 successive infants to die.
If each parent had 1 fatal recessive genetic defect, then 1/4 of their children would die, so the odds are 1/16 that two successive children would die. But actually a lot of fatal birth defects are more complicated than that simple mendelian pattern.
It's even more complicated because some mothers have been captured on video trying to smother their children.
In my case I just wrote out the possible combinations, and saw the 1/3 right away. I had fallen for the same problem as the 100 coin tosses: by the time I have identified a child enough to refer to the "other child," I have already beaten some of the odds!
But please clarify the "one is a girl" and "one is named Mary."
Evil is the money of root.
I explicitly said that they have at least one child named Mary. They could have two; it still works out.
Yes, I assume that not every girl is named Mary.
"Skill shows through where genius wears thin." -Wittgenstein || Religion: uniting aviation and architecture.
What a load of crap, back in 2006 NBC dateline had a bunch of muslims go to a Nascar race and see if they were harassed, guess what they were NOT bothered at all. This sort of idiotic bollocks is what perpetuates the myth that the US is full of racists.
I could only presume that DHS officials, and boarder guards don't hang out at Nascar events, because so-called "racial" profiling does go on in law enforcement. Racial profiling itself isn't necessarily wrong (however you may interpret that word), but it does lead to quite a lot of false positives and can perpetuate ethnic animosities. And I certainly doubt that NBC Dateline was able to read people's minds, because observing public behavior at a sporting event is not generally indicative of the types of stereotypes or prejudices people may or may not have. Journalism does often present itself as a shallow type of pseudo-science however.
It's interesting how sensitive people are to that bit of humour or rhetoric that spawned this thread. It obviously hit a nerve with people. Fascinating.
Same concept, different type of broad.
I'm actually reading this book right now: The Drunkard's Walk by Leonard Mlodinow (http://www.amazon.com/Drunkards-Walk-Randomness-Rules-Vintage/dp/0307275175/ref=sr_1_1?ie=UTF8&s=books&qid=1248279392&sr=8-1)
This is a major theme of the book, including a story about the author learning he had tested (false) positive for HIV.
So far it is a good book. Maybe not as technical as I personally might desire, but I think a great read for the general populace.
What's arabic for "WHOOSH" ?
-Billco, Fnarg.com
despite them being an optional part of the Muslim faith so most turban wearers are from entirely different religions which actually require them.
http://en.wikipedia.org/wiki/Turban
Please educate yourself before posting such drivel.
the turban is NOT part of the muslim faith...neither optional nor required.
--- widget evolution: enhanced, plus, super, ultra, extreme, exxxtreme, ultra-extreme,
You're all thinking about this the wrong way. Clearly, we need more terrorists to make the math easier to understand. Then we wouldn't have all this confusion! The boys at the TSA would be very grateful.
The probably easiest way would be to somehow force people to only report the probability conditional on the fact they are tested positively, i.e. compare P(Not being a terrorist | identified as terrorist) to P(Being a terrorist | identified as terrorist)
e.g. Instead of saying a test has 90% chance to be positive (as stated), report that the test is wrong 90% of the time.
Additionally, I wonder if there ever was any real terrorist that answered truthfully to the question "Are you a terrorist?" which they ask you when you fly to the US...
The reason there is a greater chance in the Mary case, as opposed to the girl case, is that a family with two girls is twice as likely to name a child Mary as a family with only one girl (since they have two opportunities to name a child Mary instead of one).
In the first case, if we choose 1000 couples, on average we get:
250 with Boy/Boy
250 with Boy/Girl
250 with Girl/Boy
250 with Girl/Girl
Since we can eliminate the 250 Boy-Boy couples, the odds of Girl-Girl are 250/750 or ~33%
In the second case, lets assume that 10% of girls get named Mary. In this case we have:
250 Boy/Boy, of which 0 are Mary/NotMary and 0 are NotMary/Mary
250 Boy/Girl, of which 0 are Mary/NotMary and 25 are NotMary/Mary
250 Girl/Boy, of which 25 are Mary/NotMary and 0 are NotMary/Mary
250 Girl/Girl, of which 25 are Mary/NotMary and 25 are NotMary/Mary
Therefore, the odds of a Girl/Girl couple having a Mary is 50/100, or 50%
Math has a way of warping almost anything. Take the miles per gallon rating we use in the US to tell us how efficient our cars are. Miles per gallon is actually a very misleading measurement. What we should probably use is gallons per mile, or gallons per 100 miles.
Take an example where a Range Rover gets 14 MPG, a Toyota Rav4 gets 24 mpg, and a Prius gets 46 mpg. It isn't intuitive based on the miles per gallon, but moving from the Range Rover to the Rav4 saves more fuel than moving from the Rav4 to the Prius. That is because people don't drive a fixed number of gallons, but drive (more or less) a fixed number of miles. When you look at the gallons used per 100 miles it is clear. The Range Rover uses 7.14 gallons per 100 miles, while the Rav4 uses 4.17 and the Prius 2.17. So it is clear that changing from a Range Rover to a Rav4 will save almost 3 gallons per 100 miles, while changing from a Rav4 to a Prius only saves 2 gallons per 100 miles.
The added factor in the second case is that a couple with two daughters has twice as many opportunities to name a daughter "Mary" as a couple with only one daughter does.
must be the top reason for false positives in broad screening.
thegodmovie.com - watch it
A family with two children is chosen at random from a large population.
If I tell you only that they have at least one daughter, what is the probability that both children are girls?
Most people can get that one (it's 1/3), but fail miserably on this question:
You are incorrect. Your statistic would be true if we were randomly picking family with two children until we came across one with (at least) one girl. There's a 1/3 odds there we'd pick one with two girls, and 2/3 that we'd pick one with just one.
However, that is not what you said. You said we picked the groups at random, and, hence, telling us the gender of a child tells us nothing about the other one. The genders are entirely independent of each other.
You can see how that works by imagining that the second child has not actually been born yet.
Or imagine it as coin flips. If I announce the result of one coin flip, it's not going to alter the other. If I make pairs of coin flips, and deliberately select a pair that has at least one tails in it, however, I have removed certain flips from the odds.
You actually understand this in your second example, and get the right odds, but surreally miss it in the first, despite using exactly the same example. If only girls are named Mary, saying one is named Mary is exactly identical to saying one is a girl. Your two examples are the same. You meant for your first example to be:
If we pick out two parents who have at least one girl, what are the odds that their other child is a girl?
That has the odds of 1/3, because the possibilities are M/F, F/M, and F/F.
If corporations are people, aren't stockholders guilty of slavery?
The 9/11 and 7/7 attacks etc were instigated by self proclaimed Muslims. There's no "alleged" about it, it's just a fact. Note that I don't believe that all Muslims are terrorists, or that all terrorists are Muslim, that's just stupid. But the most widely publicised and recent terrorist attacks have been strongly linked with Islam. That doesn't mean that Islam "harbours" terrorists, a lot of Muslims don't agree with these violent attacks. But it's pretty safe to say that the majority of attempted terrorist attacks within the next few years are likely to be instigated by Muslim fanatic groups who are pissed off about the whole invasion situation in Aghanistan/Iraq/Iran. The 7/7 attacks here in the UK were "carried out by 4 British Muslim men who were motivated by Britain's involvement in the Iraq War" according to Wikipedia.
If you define "terrorist attack" as "something spectacular with a body count of at least a half dozen perpetrated by people identified as Muslim or Middle Eastern", then yes, the majority of those will probably be instigated by Muslim fanatic groups.
If you were to expand the definition to include vandalism, sabotage and assassination (but only one at a time), then we might find that "terrorists" are more diverse than that. Of course, we could always say that those guys were lone wolves, acting alone, in a total vacuum, certainly not egged on by mass media accusations that their targets are mass murderers. And definitely not endorsed by any organization, certainly not by any organization that also insists they not be called "terrorists". Those would fall under the category of "isolated incidents involving deranged individuals".
I am not a crackpot.
my fav Dumb and Dumber movie quote.
Instead of talking about false positives and negatives and dependent distributions (which fly right over the head of the average joe), boil it down to the "amplification power" of the test. A random person "presumed innocent until proven guilty" has a chance of 1/3000 to be a terrorist. If you apply your 90% test, people failing it will be terrorists ~1/333 of the time. So the test as an "amplification power" of ~9x. Now everything becomes intuitive. You are looking for a 1-in-3000 needle in a haystack with an amplification power of ~9x, you now need to look for a ~1-in-333 needle in a haystack. The term "90% accuracy" doesn't appear anywhere to confuse things, and it is something everyone can easily grasp. And yes, I know, this ignores the terrorists false negatives; for that you say the test has a "miss rate" of 1/9 so about 1 in nine terrorists will slip through. These three numbers - (1) how rare what you are looking for is, (2) what's the "amplification power" of the test, and (3) what is the "miss rate", give you enough info to intuitively convey all you need to get a good feel for how effective the test really is.
... and a utility function too!
The article is confusing because it doesn't indicate the false negative rate. You basically need to know the entire confusion matrix before inferring anything. This way, you can not only calculate the accuracy and the false positive rate, but you can also calculate the false negative rate, the precision and the recall. Precision and recall are much more useful metrics than recall when it comes to tests like these.
Also, you need to know how much it really costs you to have false negatives and false positives. If you accuse someone erroneously of being a terrorist, and the only inconvenience is a few extra minutes of body search (and the humiliation) at the airport, it *might* still be worth the trouble. If on the other hand you end up sending the poor dude to jail, and he sues you for wrongful conviction, then not so much. You therefore need to have a utility function that assesses the cost of getting it right and wrong both ways (positive and negative). That's basically what is discussed in the other article (the cost of cancer screening tests), albeit in an informal way.
"In our tactical decisions, we are operating contrary to our strategic interest."
Two consecutive identical readings was also considered conclusive proof of valid air pressure data when the venturi on a cruise missile I worked on was opened. The engineers figured that this indicated the turbulence caused by opening the venturi had dissipated, and that they could therefore use the air pressure reading as the basis for an initial altitude calculation. As it turned out, in a universe this size, it was possible to take two readings from within turbulence and get the same value, so the missile calculated that it was five miles up, and did a power dive into the sand.
Good thing it was only a test.
P. Orin Zack
+ + +
Google returns 64,000,000 matches when you search for political short stories. The first one is mine. Visit http://klurgsheld.wordpress.com/ to find out why.
When the event of interest is rare, one can report F-score (or F-measure), which is the harmonic mean between recall and precision. Recall is the proportion of cases that you have found in the population in regard of the actual number of cases, precision is the proportion of the cases that you have found in regard to the number of cases that you believe to have spotted. In other words: Recall = #(right) / #(truth) Precision = #(right) / #(hypothesis) F-score = 2 * Precision * Recall / (Precision + Recall) There are variants to adjust between recall and precision: http://en.wikipedia.org/wiki/F1_score.
What a load of crap, back in 2006 NBC dateline had a bunch of muslims go to a Nascar race and see if they were harassed, guess what they were NOT bothered at all.
Funny, I can't find any record of Dateline actually running that segment. What I do find is a billion news articles about how NASCAR and others like Michelle Malkin got their panties in a twist about it with the typical faux indignation of the bigotted right. I would have expected NASCAR's PR people to be smarter than that, but apparently not.
This sort of idiotic bollocks is what perpetuates the myth that the US is full of racists.
You've picked a strawman. Just because such attacks do happen does not mean that "the US is full of racists" what it does mean is that there are some racists here. Don't pretend that just because your silly strawman is false that no such racist attacks happen at all.
http://ibnlive.in.com/news/sikh-attacked-in-another-hate-crime-in-new-york/57501-3.html
http://www.nypost.com/seven/09162007/news/regionalnews/muslim_biz_gal_beaten.htm
When information is power, privacy is freedom.
Of course not, they fit right in. Pretty much everyone at a Nascar race likes to blow stuff up.
In that case the probability will be "close to 50%" assuming Mary is a reasonably rare name amongst girls.
Assume 10% of girls are named Mary, and a sample size of 2000
Oldest child - Youngest
Boy - Boy 500
Boy - Mary 50
Boy - Girl not mary 450
Mary - Boy 50
Mary - Mary 5
Mary - Girl not mary 45
Girl not Mary - Boy 450
Girl not mary - mary 45
Girl not mary - girl not mary 405
Probability of 2 girls assuming at least one is called Mary = (5 + 45 + 45 + 405)/(2000 - 500 - 450) = 500/1050
This is still not 50% but p(2 girls) does approach 50% as p(girls name is Mary) approaches 0
That's what (some ? most ?) other countries do : in France we use liters per 100 kilometers.
However, I never thought about the advantages of it, thanks for your post !
The relevant statistical concepts are the following:
True positive (TP)- The person really is a terrorist, and your test detects him (I'm going to use the male pronoun throughout to keep the language less cluttered)
False positive (FP) - The person is not a terrorist, but your test says he is
False negative (FN) - The person is a terrorist, but your test does not detect him
True negative (TN) - The person is not a terrorist, and your test correctly says he is not
Sensitivity is the ability of the test to detect a true positive and equals TP / (TP + FN), or to put it another way, TP / (number of all terrorists, whether detected or not)
Specificity is the ability of the test to detect a true negative and equals TN / (TN + FP), or to put it another way, TN / (number of people who are not terrorists, whether the tests says they are or not)
Generally in statistics, sensitivity and specificity are considered to be properties of the test, independent of the population it is run on, although this probably isn't strictly true in the real world.
Positive predictive value (PPV) is the chance that someone who tests positive is really a terrorist and equals TP / (TP + FP), or to put it another way, TP / (all people who test positive, whether they are terrorists or not)
Negative predictive value (NPV) is the chance that someone who tests negative is really not a terrorist and equals TN / (TN + FN), or to put it another way, TN / (all people who test negative, whether they are terrorists or not)
In these terms, the statement that the OP makes can be rephrased as: Even a test with good sensitivity and specificity can have poor positive predictive value if the frequency of terrorists is low.
Example: Test is 99% sensitive and 99% specific. Terrorists are 0.01% (i.e. one in 10,000)
Consider 1,000,000 people. Terrorists: 100, not terrorists: 999,900
True Positives = 0.99 (sensitivity) * 100 (terrorists) = 99; False Negatives: 100 (terrorists) - 99 (true positives) = 1
True Negatives= 0.99 (specificity) * 999,900 (not terrorists) = 989,901; False Positives: 999,900 (not terrorists) - 989,901 (true negatives) = 9,999
Positive Predictive Value = 99 / (99 + 9,999) ~ 0.0098 = 0.98%
Sorry about the lines that consist of a period, but my <br> tags were being ignored there, and I couldn't figure out a better way to insert blank lines to make the post more readable.
How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?
It can't. This is why there's a Creationism museum in Kentucky.
The difference is simple: "at least one daughter" is not the same as "one child is a girl".
"at least one daughter, what's the probability of 2 girls" works as you illustrated.
"one child is a girl, what's the probability the other is a girl" is a simple 50%. The probability that the 2nd child is a boy or girl does not depend on the gender of the first (the probabilities are unrelated). (Also, first and second are not important for this comparison. You could switch them around and the statement would still hold true.)
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
I thought this was going to be a story about looking around for women while you're drunk.
The GP is correct in this case as they did not specify whether a particular child was a girl. All we know is that at least one is a girl.
You are correct that "telling us the gender of a child tells us nothing about the other one", but we've not been told which of the children is a girl so this this is not relevant here. All we know is that we have 3 equally likely possibilities - M/F, F/M and F/F.
So in this example, these questions mean the same thing:
A family with two children is chosen at random from a large population. If I tell you only that they have at least one daughter, what is the probability that both children are girls?
If we pick out two parents who have at least one girl, what are the odds that their other child is a girl?
What is relevant is how the questioner knows that there is at least one daughter e.g. by asking the parents "Is your eldest a girl?" or "do you have any daughters?" gives probabilities of 0.5 and 1/3 respectively, but the way the question was phrased implies the latter.
What if both girls were named Mary?
un-ALTERED reproduction and dissimination of this IMPORTANT information is ENCOURAGED
That is, however, nearly useless information.
Most child molesters are known to the child and the family.
Does this mean that we should consistently pick new people that we don't know to watch our children??
It's not that "a lot of Muslims don't agree with these violent attacks", it's more that a lot of Muslims don't understand why westerners think these attacks have anything to do with being Muslims. Timothy McVeigh was a white guy who won a Bronze Star in the Gulf War. Should we start treating white male veterans differently because some fraction of less than 0.0% of them blew up a federal building? Muslims are confused by questions such as "where's the outrage in the Muslim community about terrorism?". Their response is typically something along the lines of "uhm, what? those guys weren't Muslims, they were whackos that happened to be Muslims or claimed that they were Muslims". This is pretty much exactly like the reaction of the average Catholic in the USA while the IRA was blowing people up in Northern Ireland. "Uhm, what? they're Catholic? uh, whatever, they are a bunch of terrorists blowing people up in Northern Ireland".
Being Muslim has as much to do with terrorism as being Irish, or a military veteran.
Being a loon that thinks you can terrorize people in to doing what you want them to do is the defining characteristic of terrorists, but unfortunately there is no loon uniform, loon language or loon skin color so we tend to fall back to simple associations and that's not helpful in the slightest.
> How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?
In One Word: it can't.*
*Yes, that's two words, one of which is a contraction. Statistically, it's close enough. And I'm much better at statistics than anyone in Congress.
There was a similar case on one of the Law & Order TV shows.
Thankfully for the guy who was framed, the state's forensic expert noticed the semen didn't look right. It looked like it had been frozen. They found the real bad guy in the end of course.
Still, as a juror, if a guy accused of rape claimed he was framed I would want either a good alibi or a good explanation of how someone acquired his semen, or a statement from the victim that didn't jibe with this guy being the do-er. Absent that, framed or not, the evidence of his guilt would be "beyond a reasonable doubt."
If it was a rape-murder though, that tinge of doubt might be enough to keep me from giving the guy the needle. When it comes to the ultimate penalty, I want to be absolutely sure. Reasonable doubt just isn't good enough.
The change from 14 to 24 being a better improvement than that of 24 to 46 is immediately clear to anyone who thinks in terms of ratios rather than differences.
24/14 > 46/24
The problem isn't using mpg vs. gpm, the problem is people who don't remember their elementary-school maths, or who never learned them.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
> How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?
I'm in favor of making basic statistics, and also basic logic, required units in gradeschool. Everybody should know what standard deviation is, and statistical significance, and a premise, and the difference between soundness and validity. If something has to be cut from the curriculum to make room, I can propose a long list of significantly less important stuff: cursive handwriting, Johnny Appleseed, casting out nines, analog clock reading (who the heck can't afford a digital clock these days), the list just goes on and on.
Cut that out, or I will ship you to Norilsk in a box.
If you've paid attention to the cancer warnings in the press, they don't tell you to go see a doctor for every mole.
What they do tell you is to monitor any changes in your moles, including of course new ones. Compare them against reference pictures of what is healthy and not healthy, and use common sense to see if there is another plausible explanation. If you have a bleeding mole but you just cut yourself there, it's very unlikely to be cancer and you don't need it tested.
If your body has a history of moles changing in a particular pattern and your doctor has previously told you not to worry, new changes that fit the same pattern are unlikely anything to worry about.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
Here's an example in the opposite direction, which I think will make things a little more clear. Suppose I were to set up a "reverse lottery," which works as follows. You have, let's say, a net worth of $100,000. If you sign up for my lottery, I pay you a dollar. Then you pick six numbers between 1 and 10, I draw six balls out of urns, and if the numbers match ... I take everything you own. Your house, your car, your computer, the clothes off your back. You're turned out on the street.
This was the concept behind the original Lloyds of London insurance syndicate. "Names" pledged everything they owned to underwrite the risks of those paying the premiums.
I payed you and your fellow Names a premium based on the value of my ship and its cargo and the risk, plus a profit for you.
So did a lot of other customers.
If my ship sank, you all paid me the declared value.
If by some fluke all the ships sank at about the same time, we the insured took you to court and collected everything you all owned, your horse, your slide rule, your coat and top hat. You were turned out on the street.
Fortunately for the Names, that didn't happen until the late 20th century.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
Both of these systems are based on volume of fuel - mass of fuel would be a fairer comparison. Measuring by volume makes diesel engines appear more efficient, simply because diesel fuel is denser. More mass of diesel fuel will fit in a liter or gallon than of gasoline.
You are correct that "telling us the gender of a child tells us nothing about the other one", but we've not been told which of the children is a girl so this this is not relevant here. All we know is that we have 3 equally likely possibilities - M/F, F/M and F/F.
No. We already know the gender of one child. Either we know that the first, or the second. We don't know which one it is, but we do know we know one of them.
Ergo, we're got two options, equally likely. We're trying to find out, either F/[?] or [?]/F, we don't know which.
And hence we have four options. Either we've got F/[M] or F/[F] if we've learned the first one, or we have [F]/F or [M]/F if we've learned the second one. We don't know which to use, because we don't know which we've learned, but luckily they have the same 50/50 odds.
Just because two of the outcomes result in two daughters don't mean they're the same, anymore than F/M and M/F are the same outcome. Knowing the first child, and the second being female, has just as much odds as knowing the first child and the second being male, and the same with swapping 'first' and 'second'. Four outcomes, not three.
It's only when the original choice of the family, in some way, depended on one of the children being female that the probability changes to 1/3. Like if you go around selecting families with one daughter, which means there's a 2/3 chance the other is a son.
If corporations are people, aren't stockholders guilty of slavery?
Actually, I can make this point much simpler. You agree that if they say 'My eldest is a girl', they have a 50% chance of having another girl. Presumable that would also apply if they said 'My youngest is a girl.', right?
And when they say 'One of my children is a girl', what they are actually saying is 'Either my youngest is a girl or my oldest is a girl.', right?
As either of those possibilities would result in a 50% chance of another girl, the situation as a whole must result in a 50% chance of another girl.
If corporations are people, aren't stockholders guilty of slavery?
People play a slightly different kind of reverse lottery every day. It's called insurance. The expected value of insurance is always negative, but you are paying to reduce your negative variance. The lottery is simply paying to increase your positive variance. Expected value isn't the whole picture (and by the way, when the lottery is large enough, it can actually have a positive expected value)
How about you define terrorism as it's always been defined ? Ideology-motivated murders (as opposed to profit-motivated, or racist-motivated, or ...)
That would both be a reasonable definition and would classify ideologies as murderous like so :
1) islam, idiotically long history of incessant and senseless attacks (including over a 1000 useless attacks with the nothing but the purpose of killing on constantinople)
2) communism (but 1000x less than 1)
3) various other religions (again a factor 100 or 1000 less than the 2 previous)
The extent people are willing to go to deny the obvious truth is amazing. Islam is an ideology that motivates disproportionally many deaths, there is no reasonable argument possible that states otherwise. And yes, muslims follow islam, and can therefore at the very least be said to be "more prone to murder", simply and only due to that ideology. That's called statistics. Men are also more prone to murder.
Whether that's a causal relation can also be established. Take 2 people's who share everything, except religion. Like, for example, the population of the Kashmir valley, meaning the only difference between them is religion. And yes the muslims kill more, much, much more.
So that leaves one of 2 explanations : either islam causes people to kill others, or the killings cause people to become muslims.
But I guess any politically incorrect truth, no matter how blatantly obvious, must be fought with all force.
Great way to legalize 3rd rate citizen and race discrimination. We only hope that we are the lucky select few to be privilege first class Fascist neo-nazi-sedo-citizen. Hail Britannia Hail Britannia Hail Britannia... Geass!!! Hail Lelouch! Hail Lelouch! Hail Lelouch!
I predict that the controlling party statistic abilities will coincidently degrade by insane amount and the old British Lord system come into play. Only 2% are first class and the rest 3rd rate citizen. Kind of what we have now with Rich, middle, and poor class, but you get Rich with everything and poor slave with nothing like how it was back before the french revolution.
How can this counter-intuitive fact be communicated effectively to people unschooled in statistics?"
We could start by never claiming any test is X% accurate. Any detection scheme for anything cannot have it's accuracy defined by only one number. You need at least two numbers...the rate of false negatives (or efficiency, which is what is usually quoted), AND the rate of false positives to get a picture of what any test will be capable of...Even the example in the linked article was wrong. a 90% efficient test does not necessarily have a 10% rate of false positives. It could have a 1% rate of FP, or a 50% rate of false positives. Saying a test is 90% accurate has no meaning by itself.
You are making this more complicated than it needs to be.
Let's assume a population of 4 thousand if you like. I hope you'll agree there are 4 equally likely possibilities:
1) M - M 1000
2) M - F 1000
3) F - M 1000
4) F - F 1000
Then we apply the random selection and the further knowledge we are given, i.e. at least one of the children is a girl. This discounts M - M and we can deduce the probability of F - F is 1000/3000.
Also from the table above we can work out if we know the eldest is F then the chance of the other being a girl is 1/2 and if we know the youngest is F then the chance of the other being a girl is also 1/2.
And hence we have four options. Either we've got F/[M] or F/[F] if we've learned the first one, or we have [F]/F or [M]/F if we've learned the second one. We don't know which to use, because we don't know which we've learned, but luckily they have the same 50/50 odds.
The mistake here is you've included the F/F set twice because you are treating F/[F] and [F]/F as different sets, which they are not. Consider what you're saying here, that p([F]/F or F/[F]) is twice p(F/M) and twice p(M/F). Or another way of looking at it is the probability of the eldest being F is 75% if you consider F/[M], F/[F], [F]/F and [M]/F to be equally likely (25%), which is incorrect.
It's only when the original choice of the family, in some way, depended on one of the children being female that the probability changes to 1/3. Like if you go around selecting families with one daughter, which means there's a 2/3 chance the other is a son.
For the purpose of this probability calculation, the knowledge we have "at least one daughter" is equivalent to "randomly select from families with at least one girl". In both cases we start off with the same information on which to base the calculation. I'm interested to know why you see there's a difference.
Incidentally, if the question was phrased something like "a family was chosen at random, and I met just one of the children, who was a girl", you will get the 50% probability you mention, but this cannot be deduced from the original question . "First child I met was a girl" implies "at least one daughter" but the reverse is not true.
There was an article on the problem of people not understanding probability percentages in Scientific American a few months ago, going specifically into the HIV tests.
The proposed solution was to use more "natural frequencies" as in "out of 10,000 people one person has HIV". That is, focusing on the counts and numbers and avoiding the 99%, five nines and 0.01% failure rates like the plague due to their inherit relativeness, as in something that decreases your chances of cancer from 0.02% to 0.01% is touted - technically correct, but misleading - as reducing your risk of cancer by 50%.
The most scary thing about that article was that doctors - medical practitioners - were doing just as bad as the general populace at understanding the probabilities and false positive risks when presented as percentages. Moving to natural frequencies the docs did a lot better, probably because it is more intuitive.
http://www.scientificamerican.com/article.cfm?id=knowing-your-chances
I'd agree about the Irish, but as for the Muslim association with Al Qaeda etc, read Al Qaeda's Ideology. Al Qaeda are trying to get back to a purer for of Islam, that is the motivation for their attacks. It's not just a vague link (though no doubt some of the people attracted to Al Qaeda will be the type of people who are looking for an excuse for a bit of violence), they're literally doing what they are commanded to do in their scriptures - what other modern Muslims are too pussy-footed to do. Just as you don't see many (any?) modern Jews sacrificing pigeons, goats and cattle all the time for their sins.. it's right there in their scriptures, and they were still doing it around the time of Jesus, so why have they stopped now?
Btw you can't get a fraction of 0.0% - it's already nothing.
which is totally what she said
I don't know: fuel is sold per volume not per mass..
Unacceptable to whom? And why?
I would imagine that any of this guys family might not see the funny side of the joke:
http://en.wikipedia.org/wiki/Balbir_Singh_Sodhi
Would the average American find me joking about the World Trade Center funny just because I thought the joke was about the lunatics flying the planes? I personally think not, so I would not risk telling jokes of that nature.
I dont read
the turban is NOT part of the muslim faith...neither optional nor required.
I personally do not associate Turbans with the Muslim faith but they are mentioned on the page I linked to so I thought it best to hedge my bets a little. Thanks for the correction.
I dont read
Are you sure? I make it that the answer can be anything from 1/3 up to, but not including, 1/2 depending on what the probability is of a girl chosen at random being called Mary.
Yes, I'd considered the same thing when I heard that some countries measure efficiency in litres per 100 km.
Also notice, though, the interesting tidbits that can easily be found from the reciprocal natures of the two values:
(miles/gallon) * n gallons = How far will this fill-up take me? (or, how long until I fill up again?)
(litres/100 km) * n km = How much is this trip costing me?
Noting the fuel prices in the US vs. abroad, I find it not all too surprising that in the US we're more interested in knowing when we fill up next vs. finding out how much each drive is costing us... ;)
This slightly-flamebaitish post was intended to be mostly funny, not offensive. Sorry if you didn't appreciate my warped humour.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Worth noting, the 2nd formula gives cost in litres, which of course needs to be multiplied by, say, euros/litre to find cost in monetary units...
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
For the purpose of this probability calculation, the knowledge we have "at least one daughter" is equivalent to "randomly select from families with at least one girl". In both cases we start off with the same information on which to base the calculation. I'm interested to know why you see there's a difference.
No, it's not the same thing at all. The possible outcomes are the same, but the odds are not. (Which you should understand because, strictly speaking, M/F and F/M are the same outcome if we're not asked about which is which.)
Incidentally, if the question was phrased something like "a family was chosen at random, and I met just one of the children, who was a girl", you will get the 50% probability you mention, but this cannot be deduced from the original question . "First child I met was a girl" implies "at least one daughter" but the reverse is not true.
Because that was the original question. I quote:
A family with two children is chosen at random from a large population.
If I tell you only that they have at least one daughter, what is the probability that both children are girls?
In other words, a family was chosen at random, and we are told the gender of one of the children. We did not select a family with at least one daughter. We selected a family entirely randomly, then it turns out they have least one daughter, which of course has no bearing on anything else. We might as well be told one of their children is 5'4" and then asked the gender of the other.
Or, to put it more sanely, like I did earlier...the question is identical to stating that 'either the youngest or oldest is female'. Both of those statements have a 50/50 chance of the other being female, and if they are the only two options the thing as a whole must be 50/50. That's how probability works.
The question you think was asked, and the question the original poster thought he was asking, is not the question that was actually asked, as I was attempting to point out. The original poster thought he'd asked your question, 'If we pick a family with at least one daughter, etc', a question with odds of 1/3, but he did not ask that one.
It's like me flipping a coin twice, and then telling you one of the flips was heads. That doesn't change anything. You think that we're doing a bunch of coin flips in pairs and selecting pairs with at least one head, which, indeed, would alter what the odds of the other are...but that's not what we're doing. We're selecting a pair of flips and randomly stating what one of them was.
Now, it's entirely possible you don't agree with my parsing of the original question, which is understandable, it's not a very good question. I'm not going to debate semantics. If you want to read that it is asking something else, that they deliberately selected a family with a female child, fine.
However, do you agree that the question 'We find a family with two children. We flip a coin to pick one of them, and inform you that one is female. What are the odds the other one is also?' has a 50/50 odds? And the same odds if, on another family, instead of flipping a coin to pick, they tell you the oldest child is female?
If corporations are people, aren't stockholders guilty of slavery?
Muslims are too pussy-footed
So that's why Muslim women can't show their feet in public.
In that case, I pity them.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Turbans are worn by Sikhs. This is a completely different religion to Islam which is alleged to harbour these terrorists.
Different, yes, but not completely different. Sikhism is a combination of Islam and Hindu ideas. I don't know of any incidents of Sikh terrorism, but there are definitely a strong militaristic aspect to their teachings (which is why they're supposed to carry a dagger and wear a steel bracelet at all times).
Under capitalism man exploits man. Under communism it's the other way around.