The Language of Argument

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12

Choices

Probabilities are used not only when we determine what to believe but also when we

choose what to do. Although we sometimes assume that we know how our actions

will turn out, we often have to make decisions in the face of risk, when we do not

know what the outcomes of our options will be, but we do know the probabilities

of those outcomes. To help us assess reasoning about choices involving risk, this

chapter will explain the notions of expected monetary value and expected overall

value. Our most difficult choices arise, however, when we do not know even the

probabilities of various outcomes. Such decisions under ignorance or uncertainty

pose special problems, for which a number of rules have been proposed. Although

these rules are useful in many situations, their limitations will also be noted.

Expected Monetary Value

It is obvious that having some sense of probable outcomes is important for

running our lives. If we hear that there is a 95 percent chance of rain, this

usually provides a good enough reason to call off a picnic. But the exact

relationship between probabilities and decisions is complex and often

misunderstood.

The best way to illustrate these misunderstandings is to look at lotteries

in which the numbers are fixed and clear. A $1 bet in a lottery might make

you as much as $10 million. That sounds good. Why not take a shot at $10

million for only a dollar? Of course, there is not much chance of winning the

lottery—say, only 1 chance in 20 million—and that sounds bad. Why throw

$1 away on nothing? So we are pulled in two directions. What we want to

know is just how good the bet is. Is it, for example, better or worse than a

wager in some other lottery? To answer questions of this kind, we need to

introduce the notion of expected monetary value.

The idea of expected monetary value takes into account three features

that determine whether a bet is financially good or bad: the probability of

winning, the net amount you gain if you win, and the net amount you lose

if you lose. Suppose that on a $1 ticket there is 1 chance in 20 million of win-

ning the New York State Lottery, and you will get $10 million from the state

if you do. First, it is worth noticing that, if the state pays you $10 million,

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