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Problem of the Week 1108

A True Lottery Story

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Alice and Bob are at a conference dinner and each of them holds a ticket for a prize drawing. There are 100 prizes and 200 tickets. The prizes are given by choosing tickets randomly; once chosen, a number is not eligible for a second prize.

Alice wants to leave so that she can watch a Macalester soccer game on the webcast. So she gives her ticket to Bob, saying: "Here, this will double your chances of winning something." Bob is not so sure. If there was only one prize, then holding two tickets would indeed double his probability of winning something. But if there were 200 prizes, then the probability of winning is 1, which is not improved by holding two tickets. When there are 100 prizes, what is the ratio of Bob's chances with two tickets compared with his chances with just one ticket?

Of course, one can investigate more in this area. Replace 100 by p (for prizes) and find the functional relationship of the ratio as a function of p.

Investigate what happens if several people, as opposed to just Alice, give Bob their tickets.

Replace 200 by n.

Source: True story involving Stan Wagon and Ed Packel at a recent conference.

© Copyright 2008 Stan Wagon. Reproduced with permission.

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27 October 2008