Resolving the unfair subway

From my earlier post:

Marvin gets off work at random times between 3 and 5 P.M. His mother lives uptown, his girlfriend 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.

Downtown trains run past Marvin’s stop at, say, 3:00, 3:10, 3:20,…, etc., and uptown trains at 3:01, 3:11, 3:21,…. To go uptown, Marvin must arrive in the 1 minute interval between a downtown and an uptown train.

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2 thoughts on “Resolving the unfair subway

  1. You may consider the fact that given a true 50/50 chance scenario, it is perfectly reasonable that occasionally a sequence exists such that out of 20 trials, only 2 of them go one way and the other 18 go the other.

    Take a coin flipping. It could land HHHHH HHHHH HHHHH HHHTT, or TTHHH HHHHH HHHHH HHHHH HHHHH, or HTTHH HHHHH HHHHH HHHHH HHHHH or THTHH HHHHH HHHHH HHHHH HHHH or any such combination. While it is not likely to happen, it is perfectly possible that this is just a spike in favor of one train instead of the other and that in the next 10 days he might end up having dinner with his mother.

  2. Yes that’s definitely possible.

    One thing baffles me though: why does Marvin tell his mother that she stands a 50-50 chance of having dinner with him? If he has an equal chance of arriving at the station at any one time, shouldn’t his mother only stand a 1/9 chance of arriving between 3:00 and 3:01?

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