Gym Puzzle

It is 7pm at a gym. The manager asks the receptionist if member A is likely to turn up. The receptionist has access to a computer log and is able to generate and use spreadsheets to help answer such questions.

The receptionist has an idea that A tends to come on the same nights as B, and that B generally arrives before 7, whereas A generallly arrives after. Using a spreadsheet determines the numbers of nights they have been together or seperately. How should the receptionist reply?

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A trick question, of course! Suppose that A and B have not arrived, and that B has never been later than 7.The receptionist can easily estimate P(B|¬A) as the number of nights A and B were both present divided by the number of nights that B came. This can be interpreted as ‘the probability that B will come, based on the data used to calculate it’. But is this the appropriate data to use?

I imagine that a typical member will rarely go two nights in a row, and will not wish to miss too many nights in a row. The receptionist could use the log data just for A to determine ‘the probability that B will come, based on this log data’. They could even take initial subsets of the data and check that this gave reasonably stable results. If A hadn’t been coming long enough for this to give much confidence, they could check the method against other members’ data, hopefully confirming my conjecture.

But which is the appropriate probability? The key question is, are A’s visits significantly influenced by B’s presence? The receptionist might have an opinion about the social lives of the members, but this would rarely be conclusive. But if the gym has a bar it could be that an anlysis of reciepts might give a strong indication. (E.g., if whenever A and B are both present they alternately buy drinks for two.) But is there a general solution?

My own approach would be to quote a probability based on my conjecture (assuming that it is not disconfirmed) but to note any alternatives for which I see some evidence and the range of probabilities this yields, while noting that this depends on my understanding of the social lives of gym members, which the manager may or may not wish to discount. Maybe they have other conjectures I could analyse? Or maybe they would like to ask other staff, or other gym managers? Or academics, or … .

Dave Marsay

 

 

 

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