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A family has two sons. The probability that a son is a boy or a girl is $1/2$.

We ask the father to choose one of them and to tell us the genre. If the father says "Boy!", what is the probability that both them are boys?

My attemp is:

E=event that the father tells "Boy"

F=event that both sons are boys

$$p(F|E)=\tfrac{P(EF)}{P(E)}=\tfrac{P(F)}{P(E)}=\tfrac{\tfrac{1}{4}}{\tfrac{1}{2}}$$.

But the solution proposed was

$$p(F|E)=\tfrac{\tfrac{1}{4}}{\tfrac{1}{3}}=\tfrac{3}{4}$$ Why?

Anne
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1 Answers1

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Your answer is right.

Very likely the problem meant to ask "We ask the father to tell us the identitiy of a boy, and he does. What is the probability that the other child is a boy?". The answer to that one is $\frac13$.

The solution proposal is not right under any plausible interpretation of what the question meant to ask.

Mark Fischler
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  • Did you mean to say $1/2$? – Myridium Apr 03 '17 at 15:11
  • No, I meant $\frac13$ for that different question that I assume the questioner meant to ask. As I said in my first sentence, the answer the question that was actually presenteed is $\frac12$, as the OP surmised. – Mark Fischler Apr 03 '17 at 15:17
  • Maybe it's just me, but I don't understand "We ask the father to tell us the identitiy of a boy, and he does. What is the probability that the other child is a boy?". The identity of someone is their personal information; i.e. name etc. I really don't know what you mean by this usage. – Myridium Apr 03 '17 at 15:19
  • @Myridium I would have said 1/2 . If the father says that one of the 2 sons is a boy, the other one could be a boy or a girl . But I could consider that the sons are MF, FM or MM ...in this case 1/3 could make sense. – Anne Apr 03 '17 at 17:16