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Question: Suppose a family has 2 children, one of which is a boy. What is the probability that both children are boys?

My attempt:

P(both are boys | one is a boy)

= P(both are boys AND one is a boy) / P(one is a boy)

= P(one is a boy | both are boys) * P(both are boys) / P(one is a boy)

= 1 * (1/4) / (1 - 2(1/2)^2)

= 1/2

But the real answer is 1/3. Why?

  • 0
    Where do you get $1 - 2(1/2)^2$ in the numerator? – 2017-01-03
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    Please provide all given information (i.e. the probability of a child being a boy) – 2017-01-03
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    You mean the denominator? I assumed P(one is a boy) = one minus the probability of two boys or two girls = 1 - (1/2)^2 - (1/2)^2 = 1 - 2(1/2)^2 = 1 - 1/2 = 1/2 – 2017-01-03
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    @user7368066, yes, denominator, my bad. Note that you are using "one boy" in two different ways: in the entire computation, "one boy" means "at least one boy", but then to compute the denominator you interpret it to mean "precisely one boy". If you use the same interpretation consistently, you get the right answer. – 2017-01-03

2 Answers 2

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Your second to last line should be

$$\frac{1\cdot\frac14}{1-\left(\frac12\right)^2}$$

So you wrote one $2$ too many. There is no reason for the $2$ in (1 - 2(1/2)^2)

  • 0
    Where is my error in: P(one is a boy) = one minus the probability of two boys or two girls = 1 - (1/2)^2 - (1/2)^2 = 1 - 2(1/2)^2 = 1 - 1/2 = 1/2 – 2017-01-03
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    @user7368066 One boy means "at least one boy" (otherwise, `P(one is a boy | both are boys)=0`) . So don't subtract the probability of two boys. – 2017-01-03
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Assuming boys and girls have same probability to be born. You have the following cases for the two children: BB, BG, GB (B is boy, G ir girl). From those 3 only 1 is two boys so the probability is $1/3$ as stated.

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    I don't see why marked down by people, it seems to be an answer. – 2017-01-03
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    Mind explaining why you donwvote? – 2017-01-03
  • 0
    You answer doesn't address why the OP's attempt is incorrect. You've just given a standard answer to a question which has been asked many times on this site. – 2017-01-03