| Yes, I was posting in a rush and was being sloppy. Here's a more detailed calculation. The process involved is that the couple continues to have children until they have at least one of each gender. If we assume that at each birth there is a probability p of having a boy (as I noted in my response to btilly elsewhere, the Bayesian prior would actually be a distribution for p, not a point value, but I'll ignore that here for simplicity), then the process can be modeled as a branching tree something like this: Child #1:
boy -> p;
girl -> 1 - p Child #2:
boy - boy -> p^2;
boy - girl -> p(1 - p) : STOP;
girl - boy -> (1 - p)p : STOP;
girl - girl -> (1 - p)^2 So we have a probability of 2p(1 - p) of stopping at child 2. Child #3:
boy - boy - boy -> p^3;
boy - boy - girl -> p^2(1 - p) : STOP;
girl - girl - boy -> p(1 - p)^2 : STOP;
girl - girl - girl -> (1 - p)^3 So we have a probability of [1 - 2p(1 - p)] [p^2(1 - p) + p(1 - p)^2] of stopping at child 3 (the first factor comes from the probability that we didn't stop at child 2 above). By a similar process we can carry out the tree for as many children as we want. For the case p = 1/2, which was the case I was considering, all of these expressions for the probability of stopping at child #N (for N > 1) simplify to 1 / 2^(N - 1). So the probability of stopping at or before child #N is the sum of those probabilities from 2 to N; for N = 7 that is 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 63/64. That is close enough to 1 that I ignored cases with more than 7 children; but for a more exact calculation you could add an extra 1/64 to the denominator used to calculate the likelihood (or p-value) of the specific case that was actually observed, to allow for the cases with more than 7 children. |
The structure of what might have happened does not affect those ratios. Only what was observed does.