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February 10, 2006

Strange But True!

Q. What's that $1 lottery ticket you just purchased really worth to you — its "expected value" — BEFORE the drawing, of course, when you still might win.

A. You can figure this, and in so doing, gain a way of comparing one lottery against another. Imagine a lottery where your $1 ticket buys you a 1-in-5 chance of getting another free ticket (worth $1), plus 1 in 100 to win $5, 1 in 100,000 to win $1,000, and 1 in 10 million to win $1 million, say Jeffrey Bennett and William Briggs in "Using and Understanding Mathematics: A Quantitative Reasoning Approach." The expected value of any gamble equals the size of the prize times your probability of winning it. So what is the total expected value of your lottery ticket? Figure it this way: The free ticket has an expected value of $.20 because $1 x 1/5 = $.20; next, $5 x 1/100 = $.05; then $1,000 x 1/100,000 = $.01; and finally $1,000,000 x 1/10,000,000 = $.10. Adding all four of these together yields $.36, your ticket's total expected value. But since you paid $1, on average you can expect to lose $.64 for each ticket you buy.


 
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