This is problem 24, "The Unfair Subway", in Mosteller's Fifty Challenging Problems in Probability with Solutions
Marvin gets off work at random times between 3 and 5 P.M. His mother lives uptown, his girl friend downtown. He takes the first subway that comes in either direction and eats dinner with the one he is first delivered to. His mother complains that he never comes to see her, but he says she has a 50-50 chance. He has had dinner with her twice in the last 20 working days. Explain.
The accompanying solution says that it's because the uptown train always arrives one minute after the downtown train, which in turn arrives nine minutes after the uptown train, in this time span. So there's a nine-to-one chance that Marvin will get on the downtown train and not the uptown one.
Huh? Then what happened to the "50-50 chance" part of the problem?
The problem seemed to be posed as a probabilistic inference problem, i.e. one where the goal is to calculate: $\binom{20}{2} (0.5)^2 (1-0.5)^{18} \approx 0.00018$ but it turns out it was a statistical inference problem (one based on maximum likelihood estimates at that) that contradicts information in the problem itself.
So my question is: is this a valid problem in probability? Am I missing something that would make this a valid problem?