Chapter 5: Expectation


Overview

All men know the utility of useful things; but they do not know the utility of futility.

—Chuang-Tzu

Imagine there is a quantity about whose value Alice is uncertain, like the amount of money that she will win in the lottery. What is a fair price for Alice to pay for a lottery ticket? Of course, that depends on what is meant by "fair." One way to answer this question is to say that a fair price would be one that is equal to what Alice can expect to win if she buys the ticket. But that seems to be just replacing one undefined concept, "fairness," by another, "expectation."

Suppose that the lottery has a grand prize of $1,000,000 and a second prize of $500,000. How much can Alice expect to win? $1,000,000? That is clearly the most Alice can win, but, unless she is an incurable optimist, she does not actually expect to win it. Most likely, she will not win anything at all but, if she really expects to win nothing, then why bother buying the ticket at all?

Intuitively, the amount that Alice can expect to win depends, at least in part, on such issues as how many tickets are sold, whether or not a prize is guaranteed to be awarded, and whether she thinks the lottery is fair. (Back to fairness again) Clearly if only four tickets are sold and both the grand prize and second prize are guaranteed to be awarded, she might expect to win quite a bit of money. But how much? The notion of expectation is an attempt to make this precise.




Reasoning About Uncertainty
Reasoning about Uncertainty
ISBN: 0262582597
EAN: 2147483647
Year: 2005
Pages: 140

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