of at least one boy given both are boys = 1. P(BB) = probability of both boys = from the prior distribution. P(B) = probability of at least one being a boy, which includes cases BB and G·B = + = . Note that, although the natural assumption seems to be a probability of , so the derived value of seems low, the actual "normal" value for P(BB) is , so the is actually a bit higher. The paradox arises because the second assumption is somewhat artificial, and when describing the problem in an actual setting things get a bit sticky. Just how do we know that "at least" one is a boy? One description of the problem states that we look into a window, see only one child and it is a boy. This sounds like the same assumption. However, this one is equivalent to "sampling" the distribution (i.e. removing one child from the urn, ascertaining that it is a boy, then replacing). Let's call the statement "the sample is a boy" proposition "b". Now we have: The difference here is the P(b), which is just the probability of drawing a boy from all possible cases (i.e. without the "at least"), which is clearly . The Bayesian analysis generalizes easily to the case in which we relax the 50:50 population assumption. If we have no information about the populations then we assume a "flat prior", i.e. P(GG) = P(BB) = P(G·B) = . In this case the "at least" assumption produces the result P(BB|B) = , and the sampling assumption produces P(BB|b) = , a result also derivable from the Rule of Succession. Martingale analysis Suppose one had wagered that Mr. Smith had two boys, and received fair odds. One pays $1 and they will receive $4 if he has two boys. Their wager will increase in value as good news arrives. What evidence would make them happier about their investment? Learning that at least one child out of two is a boy, or learning that at least one child out of one is a boy? The latter is a priori less likely, and therefore better news. That is why the two answers cannot be the same. Now for the numbers. If we bet on one child and win, the value of their investment has doubled. It must double again to get to $4, so the odds are 1