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 Mark Perakh's Web Site

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Title Author Date
question on example Perakh, Mark Feb 08, 2004
Dear Steven:

Thank you for your comment. It is not new, though, and was suggested more than once before. The explanation also was given more than once before. In particular, when I was teaching statistical physics, the first few weeks usually were spent on probability theory, and more than once some students asked a question similar to yours. The reason for many people being puzzled is that the mistake is subtle and therefore understandable.
Here is the explanation: You have changed the problem by imposing an additional restriction which was absent in the original problem, namely you restricted the variations of the time interval to a small fraction of the average interval. In the original problem there is no such restriction, so the interval between buses may vary in a wide range between zero and some X which is larger than the average interval. The value of X is not restricted, but has the sole limitation: the arithmetic mean of the interval must have a certain value Y<X. If the problem were formulated in this way (and this is how it is formulated for buses) your train would depart at various times throughout the day, the actual interval between two consecutive trains varying between zero and some period of time which might be anything up to a very long period, with one and only one limitation - the AVERAGE interval must be 24 hours. For example, trains could be departing at intervals varying between zero and X hours where X can be any number, say 48 hours, or 35 hours, or 53 hours, etc, but the arithmetic mean of the interval must be some Y<X, for example Y=24 hours. If you formulated the problem in such a way, it would be analogous to the original problem with buses and the answer would be Y hours. Since your problem is formulated differently, no wonder the answer is also different (X hours). Inversely, if the original problem was formulated with the additional condition that the buses arrive every minute plus-minus just a few seconds, then it would be analogous to your problem and the answer would be similar to your case. Your confusion is one between unrestricted variations (which can exceed the mean value) and small fluctuations (which are less than the mean value). Hopefully this will clarify the matter for you.

Best wishes,
Mark Perakh
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