Robert Bartoszynski - Probability and Statistical Inference

Let us finally consider briefly the third interpretation of probability, namely as a degree of certainty, or belief, about the occurrence of an event. Most often, this probability is associated not so much with an event as with the truth of a proposition asserting the occurrence of this event.

The material of this section assumes some degree of familiarity with the concept of expectation, formally defined only in later chapters. For the sake of completeness, in the simple form needed here, this concept is defined below. In the presentation, we follow more or less the historical development, refining gradually the conceptual structures introduced. The basic concept here is that of a lottery , defined by an event, say , and two objects, say and . Such a lottery, written simply , will mean that the participant (X) in the lottery receives object if the event occurs, and receives object if the event occurs.

The second concept is that of expectation associated with the lottery , defined as

(2.13)

where and are measures of how much the objects and are “worth” to the participant. When and are sums of money (or prices of objects and ), and we put , the quantity ( 2.13) is sometimes called expected value . In cases where and are values that person X attaches to and (at a given moment), these values do not necessarily coincide with prices. We then refer to and as utilities of and , and the quantity ( 2.13) is called expected utility ( EU ). Finally, when in the latter case, the probability is the subjective assessment of likelihood of the event by X, the quantity ( 2.13) is called subjective expected utility ( SEU ).

First, it has been shown by Ramsey (1926) that the degree of certainty about the occurrence of an event (of a given person) can be measured. Consider an event , and the following choice suggested to X (whose subjective probability we want to determine). X is namely given a choice between the following two options:

1 Sure option: receive some fixed amount , which is the same as lottery , for any event .

2 A lottery option. Receive some fixed amount, say $100, if occurs, and receive nothing if does not occur, which is lottery . One should expect that if is very small, X will probably prefer the lottery. On the other hand, if is close to , X may prefer the sure option.

Therefore, there should exist an amount such that X will be indifferent between the sure option with and the lottery option. With the amount of money as a representation of its value (or utility), the expected return from the lottery equals

which, in turn, equals . Consequently, we have . Obviously, under the stated assumption that utility of money is proportional to the dollar amount, the choice of is not relevant here, and the same value for would be obtained if we choose another “base value” in the lottery option (this can be tested empirically).