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A family has two children. Given that one of the children is a boy, what is the probability that both children are boys?

I was doing this question using conditional probability formula.

Suppose, (1) is the event, that the first child is a boy, and (2) is the event that the second child is a boy.

Then the probability of the second child to be boy given that first child is a boys by formula, $P((2)|(1))=\frac{P((2) \cap (1))}{P((1))}=\frac{P((2))P((1))}{P((1))} = P((2))$ ...since second child to be boy doesn't depend on first child and vice versa. Please provide the detailed solution and correct me if I am wrong.

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marked as duplicate by David K, Lord Shark the Unknown, N. F. Taussig, Lee David Chung Lin, Arnaud D. Feb 14 at 15:31

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  • $\begingroup$ I don't understand what your answer is. Can you clarify your solution? $\endgroup$ – saulspatz Feb 13 at 19:04
  • $\begingroup$ Is the given information There is one boy or There is at least one boy? $\endgroup$ – timtfj Feb 13 at 19:11
  • $\begingroup$ What is the probability of having a boy or a girl? Are they the same? $\endgroup$ – Jonathan Perales Feb 13 at 19:20
  • $\begingroup$ What you've written is not clear. Are we told that the children are numbered somehow (by age perhaps?) and that the first one (eldest) is a boy? That's not the same assumption as saying that "one of them is a boy". $\endgroup$ – lulu Feb 13 at 19:34
  • $\begingroup$ This is a famously ambiguous brain teaser. It has been analyzed to death (figuratively speaking) in answers to multiple previous questions on math.stackexchange. If "first child" means "first one born" in your answer, then your answer is correct assuming that the "given" statement always refers to the first-born child. If "first child" means "the child I was told about" then the answer depends on why you were told about that child. $\endgroup$ – David K Feb 13 at 19:44
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Let's write the sample space:

BB, BG, GB, GG

But we know that one child is a boy, so that means that GG isn't a possibility. Thus, the sample space is reduced to: BB, BG, GB.

Therefore, the probability of both children being boys given that one is a boy is 1/3.

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The flaw in your solution is to write $$P(2,1)=P(2).P(1)$$ What would be the logic behind this? Think again, the event $(2,1)$ means "there are $2$ boys and there is at least $1$ boy". Therefore, $(2,1)$ is equal to the event $(2):$ "there are two boys".

Can you fix your solution yourself now?

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