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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.

Chain Markov
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Shubham Kumar
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  • I don't understand what your answer is. Can you clarify your solution? – saulspatz Feb 13 '19 at 19:04
  • Is the given information There is one boy or There is at least one boy? – timtfj Feb 13 '19 at 19:11
  • What is the probability of having a boy or a girl? Are they the same? – Jonathan Perales Feb 13 '19 at 19:20
  • 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". – lulu Feb 13 '19 at 19:34
  • 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. – David K Feb 13 '19 at 19:44
  • To see what I mean about "analyzed to death", type "one is a boy" in the search window at the top of the page and look at the results you get. – David K Feb 13 '19 at 19:45
  • The wording is ambiguous. If it means precisely "there is at least one boy" then the probability of two boys is $1/3$ as shown below. If it means "a randomly chosen child turned out to be a boy" then the prob. of 2 boys is $1/2$. – Ned Feb 13 '19 at 22:03

2 Answers2

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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?

Arnaud Mortier
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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.

NicNic8
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