| The problem really is almost one of metagaming. People who are reading this are likely to have seen, for example, questions which read as if they're asking for a conditional probability ("John is male, 33 years old, and has a degree in English literature; what is the probability he works as a barista?") but are designed to let the questioner turn around and say "Ah-HA! I got you! It was really a question about the base rate (in this case, of baristas)!". As posed and with knowledge of that issue, this question reads like an attempt to do the opposite: to pose a question which seems like it's asking about the base rate of boys vs. girls, but then the questioner turns around with "Ah-HA! I got you! It was really a question about the conditional probability!" Once it's phrased in a way that makes explicit that it really is a question about conditional probability, and not an attempt to lure someone into a base-rate trap, there's no paradox. Complicating things is that analyses usually focus on the day of the week as the crucial factor, when it's easier to get to an intuitive understanding of the probability via dealing with the day-of-week first and then focusing on the small but crucial change that comes from knowing the gender of one of the children. After accounting for day-of-week you are left with 28 equally-probable situations, with at least one girl in 14 of them, for the expected 1/2. Then the fact that you end up at a probability just over 1/2 is due to the elimination of the case in which both children are girls (since we know at least one is a boy), which pushes the final result to 13/27 in favor of the second child being a boy. |