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waterchan
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On this website (http://www.sciencenews.org/view/generic/id/60598/title/When_intuition_and_math_probably_look_wrong), I recently came across a twist on the Two Children Problem. The problem is deceptively simple: "I have two children, one of whom is a boy born on Tuesday. What is the probability that I have two boys?"
Intuitively the answer would appear to be 1/2, but in fact it turns out that the probability is 13/27. I don't really understand the graphical solution presented on that website, so here's my own understanding of the "official" solution.
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Let's name the two children A and B. One of them is the Tuesday-born son. We don't know which one, so we consider the case that A is the Tuesday-born son. Then we consider the case that B is the Tuesday-born son.
If A is the Tuesday-born son, then B is either a boy born on any of the days (7 outcomes), or a girl born on any of the days (7 outcomes). 7 + 7 = 14 possible outcomes so far.
If A is not the Tuesday-born son, then B MUST be the Tuesday-born son. Then what is A? A can be a girl born on any of the days (7 outcomes). Or, A can also be a boy born on any of the days EXCEPT TUESDAY. Because otherwise, that would contradict the assumption for this case that A is not the Tuesday-born son. Thus, we remove the Tuesday possibility and are left with 7 - 1 = 6 outcomes.
The total number of outcomes is thus 7 + 7 + 7 + 6 = 27. The total number of desired outcomes, where both children are boys, is 7 (A is Tuesday boy, B is boy) + 6 (B must be Tuesday-boy and A is a boy) = 13 outcomes.
Thus, probability = 13/27.
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Now, personally I suspect the solution is flawed because it considers the birthday which is arguably irrelevant, and applies conditional probability to unconditional outcomes. But I am unable to prove that it's flawed.
Another intuitive reason I believe the method is flawed is the following. Consider replacing the verb "have" in the question with any other verb. For example:
I kidnapped two children, one of whom is a boy born on Tuesday. What is the probability that I kidnapped two boys? (Does the birthday and sex of the kidnapped child affect anything?)
I knocked down two children, one of whom is a boy born on Tuesday. What is the probability that I knocked down two boys? (Does the birthday and sex of the knocked-down child affect anything?)
I ran over two children, one of whom is a boy born on Tuesday. What is the probability that I ran over two boys? (Does the birthday and sex of the injured child affect anything?)
I vaporized two children, one of whom is a boy born on Tuesday. What is the probability that I vaporized two boys? (Does the birthday and sex of the vaporized child affect anything?)
For all three of these questions, the exact same calculation can be applied to arrive at the exact same answer of 13/27 (which is slightly less than 1/2, the answer obtained if the information about being born on a Tuesday is disregarded). Doesn't it seem wrong that the sex and birthday of the kidnapped/knocked-down/ran-over /vaporized child has any affect on the other's sex and birthday?
Intuitively the answer would appear to be 1/2, but in fact it turns out that the probability is 13/27. I don't really understand the graphical solution presented on that website, so here's my own understanding of the "official" solution.
_____________________________________________________
Let's name the two children A and B. One of them is the Tuesday-born son. We don't know which one, so we consider the case that A is the Tuesday-born son. Then we consider the case that B is the Tuesday-born son.
If A is the Tuesday-born son, then B is either a boy born on any of the days (7 outcomes), or a girl born on any of the days (7 outcomes). 7 + 7 = 14 possible outcomes so far.
If A is not the Tuesday-born son, then B MUST be the Tuesday-born son. Then what is A? A can be a girl born on any of the days (7 outcomes). Or, A can also be a boy born on any of the days EXCEPT TUESDAY. Because otherwise, that would contradict the assumption for this case that A is not the Tuesday-born son. Thus, we remove the Tuesday possibility and are left with 7 - 1 = 6 outcomes.
The total number of outcomes is thus 7 + 7 + 7 + 6 = 27. The total number of desired outcomes, where both children are boys, is 7 (A is Tuesday boy, B is boy) + 6 (B must be Tuesday-boy and A is a boy) = 13 outcomes.
Thus, probability = 13/27.
______________________________________________________
Now, personally I suspect the solution is flawed because it considers the birthday which is arguably irrelevant, and applies conditional probability to unconditional outcomes. But I am unable to prove that it's flawed.
Another intuitive reason I believe the method is flawed is the following. Consider replacing the verb "have" in the question with any other verb. For example:
I kidnapped two children, one of whom is a boy born on Tuesday. What is the probability that I kidnapped two boys? (Does the birthday and sex of the kidnapped child affect anything?)
I knocked down two children, one of whom is a boy born on Tuesday. What is the probability that I knocked down two boys? (Does the birthday and sex of the knocked-down child affect anything?)
I ran over two children, one of whom is a boy born on Tuesday. What is the probability that I ran over two boys? (Does the birthday and sex of the injured child affect anything?)
I vaporized two children, one of whom is a boy born on Tuesday. What is the probability that I vaporized two boys? (Does the birthday and sex of the vaporized child affect anything?)
For all three of these questions, the exact same calculation can be applied to arrive at the exact same answer of 13/27 (which is slightly less than 1/2, the answer obtained if the information about being born on a Tuesday is disregarded). Doesn't it seem wrong that the sex and birthday of the kidnapped/knocked-down/ran-over /vaporized child has any affect on the other's sex and birthday?
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