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If 3% of the population is unvaccinated, and another 3% has vaccines that don't work, I would by probability expect about 50% of the sick cases to be among vaccinated people. It looks like the vaccines are working.

You may be committing base rate fallacy: An Intuitive (and Short) Explanation of Bayes’ Theorem

The vax gets credit for all negative cases (including unvaxxed)? Clearly there are positive cases for those who were vaxxed; that is the vax did not work for them.

Some people have natural immunity, and dont need a vaccine. Some people are asymptomatic carriers - they carry a virus, but it doesnt make them sick. Some people have defective immune systems, and vaccines dont work. Vaccines operate at population level, not individual.

— Mactator (@MactatorMaximus) February 1, 2019

Good point. Then again modern CPU and GPU's can't predict 5 seconds into the future, the whole error rate, hit, miss and branch prediction is kinda staggering at just how much overcompensation actually happens. The very best is a 'near guess' and then it going down from there. Similar to the ($1/\sqrt{x}$) aka Quake's inverse square root. A near guess, and fast dirty refinements. Even then, unpredictable things happen.

why equal pay isn't really equitable

> I haven't ever heard of someone manipulating the bits in a floating point number.

That's because you lack experience. A classic trick is to use an union:

union int_float_t
{
        uint32_t i;
        float f;
};

If you actually understood the IEEE754 32-bit and 64-bit format you wouldn't be so naive.

https://en.wikipedia.org/wiki/...

Sign bit: 1 bit
Exponent width: 8 bits
Significand precision: 24 bits (23 explicitly stored)

https://en.wikipedia.org/wiki/...

Sign bit: 1 bit
Exponent: 11 bits
Significand precision: 53 bits (52 explicitly stored)

... it's Math education that needs to get better!

This seems to be a good starting point :
http://betterexplained.com/
(Something similar to his "intuitive" explanation of the exponential function and of imaginary numbers should be taught in all schools!)

But then, we have to ask ourselves, how can a teacher properly teach (the most important aspect of which is to transmit the passion for learning) a class of 30+ children (or worse, teenagers)?

There is no point in switching at the start of the exercise without any extra knowledge.

He's clearly strongly alluding to the Monty Hall problem/, where there is a significant advantage to switching. In that case, there *is* extra knowledge, but it's obfuscated somewhat --the door that Monty opens is always one of the goats, so he knows where at least one of the goats is.

That said ... I think he screwed it up somehow, because I don't see a point in switching (or in not switching) in the first place either, yet he seems assured that there is one. In the Monty Hall problem there's some extra information provided before you switch ... but not here.

All in all, I see that he's portraying it as a shell game (though the classic Monty Hall game isn't one) but something doesn't add up ...

"I know I have a limited understanding, and I do understand that git allows everyone to have a complete copy of the software."

This is where you should have stopped. You have no understanding of git. You need to learn the difference between distributed SCM and the old centralized approach.

Slashdot ate his code. It's supposed to reinterpret_cast x as an integer. Same goes for the second cast, where it should reinterpret_cast i as a float. See http://betterexplained.com/articles/understanding-quakes-fast-inverse-square-root/ for an explanation.

Some no cost resources that I don't think were mentioned:

A three part self-test/review of fundamental Calculus skills. The first 6 questions in the part on Trig, Logs and Limits are prerequisites to a first course in elementary calculus.

A collection of articles with intuitive explanations of math concepts many people find too abstract.

A textbook, "Elementary Calculus: An Infinitesimal Approach", available under a creative commons by-nc-sa license. Covers Calculus I/II material.

A collection of texts covering a sequence in Real Analysis (covers calculus concepts from an analytical point of view) and Number Theory available under a free of charge license to students using it for self-study. Probably beyond your current interest in math.

I believe the probability being halved has something to do with the birthday paradox. It's been a time where I could explain this better; if you wish to find out just search for it on Google... This page seems to have a good explanation too:

http://betterexplained.com/articles/understanding-the-birthday-paradox/

Sure, if you're calculating the chance that one specific file will match one specific MD5 sum, the chances are astronomical. But if you're calculating the chance that any of the tens of thousands of files on your PC will match any of the millions of MD5 sums in their database, the chances of a collision are much much higher. This is the birthday paradox in action.

I consider math to be software for your brain. The Romans had "Roman numerals I", but most people today have "Decimals 2.0" and "Algebra XP" installed.

Were Roman numerals invented or discovered? Were decimals invented or discovered? Both were invented, but they point at the underlying concepts that were discovered.

"e" is an human invention that points at the discovery that all growth rates share a fundamental base. "pi" is a human label for the discovery that all circles share a common ratio.

We invent the language of math (mental software) to describe relationships that already exist in the wild. We write physical equations (F=ma) to describe relationships that already exist in nature. So I think it's a little of both.

Does it matter? Yes! Not in the "practical" sense, just like art, literature, music, rainbows and *gasp* Star Wars don't matter in the practical sense. But they are still interesting in their own right, and we are curious creatures. Life would be a miserable place if it was only about making better widgets.

The feeling of invention or discovery can lead us to a better understanding of knowledge and our relationship with the world. Does Godel's incompleteness theorem change anything in your daily life? Not really, math keeps on trucking along, and 1 + 1 = 2 still works. But it's quite interesting philosophically, that some truths will always be unprovable.

I agree with the sentiment -- learning *how* to learn and collecting those aha insights is the goal of education. But it depends so much on the professor to present material in this way -- I went to a good school and the difference in teaching quality between professors for the same subject was astounding. Academia rewards publications, not the praise of students.

Shameless plug, but I've been blogging at betterexplained.com with the goal of sharing the "aha" insights that make math & science enjoyable. Given your viewpoint, you might find it interesting.

I posted this on reddit which broke the story earlier, and on my blog. Thought you might find it useful.

Quick follow-up. On digg someone posted the un-obfuscated code: http://www.cc.gatech.edu/~achille/contacts-source. txt

How it works

The code is pretty straightforward. Basically, Google docs has an embedded script that will run a callback function, passing the function your contact list as an object. The embedded script presumably checks a cookie to ensure you are logged into a Google account before handing over the list.

Unfortunately, the script doesnt check what page is making the request. So, if you are logged in on window 1, window 2 (an evil site) can make the function call. Since you are logged in somewhere, the cookie is valid and the request goes through.

Also, if you check the object that is returned, you see fields for the contact's name, email and "affinity". Presumably, a higher affinity means a more-emailed contact, so it may be possible to know the relative weight of links.

Possible solutions

Google is run by smart people and I'm sure they'll have this fixed soon. A few suggestions appear to be popping up, all centered on making sure the user is on a Google.com page and not a random site:

Referrer blocking: Block all requests from sites not in the google.com domain. However, some people run referrer-blocking software. It may be the price they have to pay for security, but there could be other consequences.

Script checks: An idea I had was to check the window.location (just like you check the cookie) to make sure it's coming from a google.com domain. This is another way to see what page is making the request.

Challenge-response: Google pages (like Gmail) can have some token or unique, computed data that they submit with their requests. Random pages won't have access to this token when they make the function call.

(From user JRF on reddit): Include part of cookie in the request URL as a unique token that only a "real" Google page would know. Need to watch out for proxies/browser history (accessible from other pages) being able to access this unique data. May need to seed or salt it in a challenge-response system.

It's interesting thinking of fixes for this - do you have any other suggestions for how Google would fix this?