COUNTERINTUITIVE PROBABILITIES REDUX
An isolated island is ruled by a dictator. Every family on the island has two children. Each child is equally likely a boy or a girl. The dictator has decreed that each first-born girl (if any) in the family should bear the name Mary Ann (the name of the beloved mother-in-law of the dictator). Two siblings never have the same name. You are told that a randomly chosen family that is unknown to you has a girl named Mary Ann. What is the probability that this family has two girls?
The dictator has passed away. His son, a womanizer, has changed the rules. For each first-born girl in the family a name must be chosen at random from 10 specific names including the name Mary Ann, while for each second-born girl in the family a name must be randomly chosen from the remaining 9 names. What is now the probability that a randomly chosen family has two girls when you are told that this family has a girl named Mary Ann? Can you intuitively explain why this probability is not the same as the previous probability?
If you need a hint, he adds this postscript:
P.S. As you know, the wording in this kind of problems is crucial. I found that the best approach to attack this kind of problems is to use Bayes’ rule in odds form. This specific form of Bayes forces you to make transparent the assumption you are (implicitly) making in solving the problem. I take the liberty to mention that in the recent third edition of my book Understanding Probability (Cambridge University Press, 2012), I advocate the use of Bayes’ rule in odds form (and Bayesian thinking in general).
Who can solve it first?