Finite Sample Description Spaces With Equally Likely Descriptions

In many probability situations in which finite sample description spaces arise it may be assumed that all descriptions are equally likely; that is, all descriptions in have equal probability of occurring. More precisely, we define the sample description space as having equally likely descriptions if all the single-member events on have equal probabilities, so that 

It should be clear that each of the single-member events has probability , since there are such events, each of which has equal probability, and the sum of their probabilities must equal 1, the probability of the certain event.

The computation of the probability of an event, defined on a sample description space with equally likely descriptions, can be reduced to the computation of the size of the event. By (6.1) , the probability of is equal to , multiplied by the number of descriptions in . In other words, the probability of is equal to the ratio of the size of to the size of . If, for a set of finite size, we let denote the size of (the number of members of ), then the foregoing conclusions can be summed up in a basic formula:

Formula for Calculating the Probabilities of Events When the Sample Description Space Is Finite and All Descriptions Are Equally Likely : For any event on

This formula can be stated in words. If an event is defined as a subset of a finite sample description space, whose descriptions are all equally likely, then the probability of the event is the ratio of the number of descriptions belonging to it to the total number of descriptions. This statement may be regarded as a precise formulation of the classical “equal-likelihood” definition of the probability of an event, first explicitly formulated by Laplace in 1812.

The Laplacean “Equal-Likelihood” Definition of the Probability of a Random Event . The probability of a random event is the ratio of the number of cases favoring it to the number of all possible cases, when nothing leads us to believe that one of these cases ought to occur rather than the others. This renders them, for us, equally possible.

In view of (7.2), one sees that in adopting the axiomatic definition of probability given in section 5 one does not thereby reject the Laplacean definition of probability. Rather, the Laplacean definition is a special case of the axiomatic definition, corresponding to the case in which the sample description space is finite and the probability distribution on the sample description space is a uniform one. This is an alternate way of saying that all descriptions are equally likely.

We may now state a mathematical model for the experiment of drawing a ball from an urn containing six balls, numbered 1 to 6, of which balls one to four are colored white and the remaining two balls are nonwhite. For the sample description space of the experiment we take . The event that the ball drawn is white is then given as a subset of by . To compute the probability of , we must adopt a probability function on . If we assume that the descriptions in are equally likely, then is determined by (7.2), and . On the other hand, we may specify a different probability function , specified on the single-member events of :

Then the function is determined by (6.1) , and .

We have thus stated two different mathematical models for the experiment of drawing a ball from an urn. Only the results of actual experiments can decide which of the two models is realistic. However, as we study the properties of various models in the course of this book, theoretical grounds will appear for preferring some kinds of models over others.

Exercises