A Rejection of Sklansky

[PairTheBoard]

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The answer is 1/3. Once we know that one child in the family is a girl, the chance the family has two girls is 1/3 or about 33.3%

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Are you saying that after the first girl is born in anticipation of the second child, you will give me 2/1 odds on my choice of the second child as a girl?


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No. Notice I said, The family has two children. One of them is a girl. I did not say the First Born child is a girl which is what you are saying when you assume the condition, "after the first girl is born in anticipation of the second child".

If all we know is that One Child is a girl the family can be one of three things, (g,b),(b,g),(g,g). Those three things Are equally likely. But only the (g,g) satisfies "the other child is a girl too".

So the Condition I gave you is that "one of the children is a girl". That is what must be conditioned on to answer the question posed in the problem.

You are now presenting a different Condition. "after the first girl is born in anticipation of the second child". That Condition cuts the possible outcomes down to two rather than the three of the problem's Condition. Your Condition cuts the outcomes down to (g,b),(g,g). You are right to say that Given Your Condition the probability the other child (which your condition defines to be the second child) is a girl, is 50%. But your Condition is different than the Problem's Condition. No fair complaining that they give different answers.

I agree there is something psychologically disturbing going on here. I've had years of training in mathematical probability, yet the first time I saw this problem I too experienced an unsettling feeling that something didn't seem right. I think it has something to do with our strong sense that girls happen with 50% probability.

Well, they do and they don't. Girls don't "happen" with 50% probability if you define Events in such a way that they don't, as in this problem. You can define lots of Events where girls don't happen with 50% probability. For example, go into the Bellagio and pick a random poker player. What is the probability your random pick will be a girl?

There's also some kind of psychology going on with the logical difference between the statements, "One child is a girl" and "The first child is a girl". I think we psychologically identify those two statements as being equivalent because for many purposes they serve about equally well for making inferences. In this case they carry very different implications relating to the question the problem asks us to answer.

Edit: The Psychology may also relate to our use of Symmetry when making inferences. And also our frequent ineptness when intuitively breaking things down into cases. We hear, "One child is a girl". We think, well if the first child is a girl it's 50-50 the other one is. And if the second child is a girl it's 50-50 the other one is. That covers all cases and by symmetry 50-50 must hold even if all we are told is that "one child is a girl". This is the kind of thinking that you will learn to unlearn when you study and practice more mathematics.

PairTheBoard