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Digital Dice by Paul J. Nahin

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17. Waiting for Buses

Our randomly arriving bus rider must obviously arrive at the bus stop between some two consecutive hours, and we lose no generality by labeling those two hours as hour 0 and hour 1. The given theoretical value of one-half hour for the average waiting time until a bus arrives, in the n = 1 case, immediately follows from the observation that the possible waiting times vary from 0 to 1 (hour), and the average of such a random quantity (with a uniform distribution) is 1/2. We can compute the theoretical value for the n = 2 case ...

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