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The Tuesday Birthday Problem
An anonymous reader sends in a mathematical puzzle introduced at the recent Gathering 4 Gardner, a convention of mathematicians, magicians, and puzzle enthusiasts held biannually in Atlanta. The Tuesday Birthday Problem is simply stated, but tends to mislead both intuitive and mathematically informed guesses. "I have two children, one of whom is a boy born on a Tuesday. What's the probability that my other child is a boy?" The submitter adds, "Believe it or not, the Tuesday thing is relevant. Well, sort of. It's ambiguous."
The problem doesn't disallow twins as it doesn't give TIME of birth. Only day. A child born at 11:50PM on Tuesday and one born at 00:15 on Wednesday are still both twins. It also does not include if in that scenario is a single egg birth or whatnot so it could still be a boy and girl twin situation.
This is related to the Principle of Restricted Choice often seen in Contract Bridge.
If the parent has two boys born on a Tuesday, he could equally have declared the other boy as being born on a Tuesday. In a parallel universe, the other boy would have been declared as being born on a Tuesday, whereas if only one of the child was a boy born on Tuesday nothing would have changed in any of the other parallel universes. Therefore the effect is the probability of 2 boys borne on Tuesday has been halved, resulting in 13/27 probability of the second child being a boy.
"I have two children, one of whom is a boy. What's the probability that my other child is a boy?" ... it is given that the FIRST child is a boy.
I must admit that English is not my native tongue but I fail to see how this gives that the FIRST child is a boy. Doesn't "one of whom" implies that it can be either the first or the second?
This is a question written in purposely misleading English.
This, in other words, is a shit question.
For large sets, this will be our guide even unto death, for the LORD will work for each type of data it is applied to...
Normal English strongly implies it does. If you say to someone I have several pieces of fruit and one of them is a banana, when in fact two of them are bananas, most people would call that lying. One could argue that strictly speaking the statement was true: you did have one banana, you just also had an additional banana, but that level of honesty is only tolerated in politicians. If you had said "at least one of which is a banana" that would be fine, otherwise the statement is deliberately misleading.
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Not necessarily. Any reading of an English sentence is an exercise in interpretation. We don't just read the words alone, we actually interpret them and use a mental model to help dismiss obviously incorrect ambiguities.
Let's say the sentence has several interpretations. For each interpretation, we could solve for an interpreted probability answer. Then we could look at each answer and ask if it makes sense. If that answer doesn't make sense, we could dismiss that particular interpretation of the sentence. If a single answer remains after that, it would be THE answer (Sherlock Holmes style).
In the example, here are some possible disambiguations:
1) "I have two children, (exactly) one of whom is a boy born on a Tuesday."
2) "I have two children, (at least) one of whom is a boy born on a Tuesday."
3) "I have two children, one of whom is a boy born on a (particular) Tuesday."
4) "I have two children, one of whom is a boy born on a (generic) Tuesday."
There is also an ambiguity in the second sentence, which is only obvious to statisticians and probabilists:
a) "What's the (Bayesian subjective) probability that my other child is a boy?"
b) "What's the (objective) probability that my other child is a boy?"
In case a), the problem is underspecified as it requires the full set of personal beliefs of the reader to be used for an answer (Bayesian subjectivists propose that a probability is merely a degree of personal belief, such that two people will not agree on the probability for the same event, because, being different people, they have different prior beliefs.)
In case b), the problem can (should) be solved solely from the problem and general common knowledge of the world (which is still required to interpret the question).
The constraints are not defined.
"I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?"
I have three children. But I also have 2 children. See the problem? I have a son born on a Tuesday, and I have another son born on a Tuesday. See the problem?
It doesn't say I have *only* two children. It doesn't say the other child can't be a son born on a Tuesday. It assumes the birth rate is 50/50, but most statistics agree it's not even. FTA, it assumes there's no such thing as twins. It assumes you have only one wife. But none of this shit is specified.
Pisses me off. Use coins and cards. Not assumed biblical customs.
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Yes, it is like the Monty Hall problem. What is the probability that 2 children are both boys? 25%. Knowing that one of the children is a boy does not change that probability as much as might be thought. The answer then is 33.3%, not 50%. This is because the additional information has cleverly NOT specified which child is the boy. If a particular child is picked out, eg. the first child is a boy, then it is 50% the other is a boy, because it always was 50% likely that a child is a boy. The bit about "born on Tuesday" does matter, because it comes close to specifying a particular child. The more improbable it is that both children fit some criteria, the closer the probability gets to 50%. If the info had been "one of whom is a boy born on Feb 29", the answer would be nearly 50%.
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That's incorrect - you've just skewed the population!
;-)
In 1000 pairs of children you'll have 250 girl/girl, 250 boy/boy, 500 girl/boy.
Of those, the ones that have at least one boy are the 250 boy/boy and 500 girl/boy pairs. So there's a 33% chance it's boy/boy if you know one is a boy.
The whole point is you could be talking about either of the boys in the 250 boy/boy pairs - it doesn't increase the probability that it's boy/boy instead of girl/boy (you're still twice as likely to have a girl/boy pair relative to a boy/boy pair). If you specify more about the boy you're talking about - for example (ironically) saying his name is Peter - then the boys are no longer interchangable and the probability tends towards 1/2.
It is tricky
My older brother and I were both born on Tuesdays.
"I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?"
As always, the challenge is the assumptions intentionally hidden in the problem statement.
"I" - was your family chosen at random, and if so, from what set?
"two children" - exactly or at least?
"one of whom" - exactly or at least?
"son" - was the sex to say chosen at random, or did you pick a child and announce his/her sex?
"Tuesday" - was the day chosen at random, or did you pick a child and announce his.her birthday?
"What is the probability..." - Some parent you are! Don't you know the sex of your own children?
Simply and honestly reveal the assumptions and the math is straightforward.
"Given a family, chosen at random from the set of all families that have exactly two children and have at least one son born on a Tuesday, what is the probability that both children are boys?"
To make the math easier, let's start with 196 families with two children, with the expected mix of boys and girls. 49 (25%) have two boys and 98 (50%) have a boy and a girl. Of the 98 boy-girl families, 84 do not have a Tuesday-Boy, leaving 14 that do. Of the 49 boy-boy families, 36 do not have a Tuesday-Boy, leaving 13 that do. That leaves a total of 27 families, of which 13 have at least one son born on a Tuesday.
So the probability is 13/27.
Reveal different assumptions, and the answer changes.
These kind of problems require competence in both English and Maths which is why so few people get them right.
Mostly they require competency in psychology, so you can figure out how the twit posing the problem is deliberately trying to mislead you by using ambiguous English and claiming on the basis of their poor communication skills to be clever.
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In my experience, many non-intutive probabilty results are easier to understand if you spell out the full population. For example, I coudn't understand http://en.wikipedia.org/wiki/Berkson's_paradox until I drawed it up graphically.