Not an easy one.1 The answer depends on the way you clarify the underlying process. Is the process that produces the data equivalent to “two tosses of a fair coin”? If so, what is the equivalent of the information we have in the present problem in the coin-tossing experiment? Is it that we know there is at least one head? If so, then the answer is 1/3, else if the information is, say, the first toss is H, then the answer is 1/2.

“Two tosses of a fair coin” is a reasonable model. When a couple has exactly two children, there are two objects \(C_1\) and \(C_2\). Don’t think that the ordering indicated by the subscripts is necessarily on the temporal scale; any two or more non-identical objects can be ordered in some way. The information given in the question says that either \(C_1\) or \(C_2\) is a boy. Here, the word “sibling” is critical, as the word does not imply any ordering relation. As the possiblities of mapping \(C_1\) and \(C_2\) to biological genders is BB, BG, GB, GG and the information given in the question rules out GG, the probability of BB is 1/3.

If the question said that the king has an elder sibling, then the answer would be 1/2, because the information would this time rule out the two possibilities GB and GG.

  1. See this Wikipedia article for further discussion. ↩