5
$\begingroup$

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?

  • 11
    The problem doesn't say there's a 50-50 chance; it says that Marvin says there is :-)2011-05-18
  • 0
    Another aspect to consider is that around 5pm there will probably be more trains going uptown than going downtown.2011-05-18
  • 2
    @joriki Yes, I was aware that the 50-50 was an indirect quote. My basic issue is that this problem goes against my notion of what a "math problem" is. If it were a puzzle or riddle, I wouldn't mind weaselly bits like that; but the fact that I have to doubt the truth of a statement (even in an indirect quote) in a math problem is what gets my goat.2011-05-18
  • 1
    This is also problem 6 in Chapter Three, Nine Problems, in Martin Gardner's first column collection, The Scientific American book of Mathematical Puzzles and Diversions, which dates it to somewhere between 1956 and 1958. Dunno whether Mosteller got it from Gardner, or vice versa, or both got it from somewhere else.2011-05-19

3 Answers 3