Specifically, I was reading this article, which discusses this wording of the question:
Consider a family with two children. Given that one of the children is a boy, what is the probability that both children are boys?
and says
In this formulation the ambiguity is most obviously present, because it is not clear whether we are allowed to assume that a specific child is a boy, leaving the other child uncertain, or whether it should be interpreted in the same way as "at least one boy".
It's not clear to me what, "specific child" is meant to mean in this context.
My thought was that it referred to calculating the probability that "exactly one" child is a boy, as opposed to the "at least one child is a boy" interpretation, but that can't be right, since, if we're given that exactly one child is a boy and we know the first one is a boy, the probability the second one is a boy would be zero. But the actual "competing" (for lack of a better word) probabilities are 1/2 and 1/3.
Additionally, if I try to come up with an equivalent problem in terms of coin flips I get still different probabilities.
e.g., "What's the probability the FIRST coin in a sequence of 2 coin flips is heads" (clearly 1/2) vs asking "What's the probability at least one of the coins in the sequence is sequence is heads?" (P(first coin is heads) + P(second coin is heads) – P(both are heads) = 1/2 + 1/2 – 1/4 = 3/4)
Where am I going wrong in my thinking here and how do I think about it correctly?
Best Answer
"Given that one is a boy" is rather confusing wording, I've seen many more people confused over the question than over the answer.
Suppose that the two children are assigned labels: Child 1, and Child 2.
If we say a specific child is a boy, then we would say, for example, that Child 1 is a boy. The question is then whether Child 2 is a boy, given that Child 1 is a boy.
If we say that (at least) one child is a boy, then we would say that either Child 1 is a boy, or Child 2 is a boy (or both), but we don't know which. The question is then whether both children are boys, given that at least one of them is.
In terms of coins, this is the difference between $\mathbb{P}(\textrm{coin 2 is heads} \,|\, \textrm{coin 1 is heads})$ and $\mathbb{P}(\textrm{coin 1 is heads} \cap \textrm{coin 2 is heads} \,|\, \textrm{coin 1 is heads} \cup \textrm{coin 2 is heads})$.