Discrete Mathematics for Computer Science

480 CHAPTER 8 Discrete Probability

If E = 0, there are no terms in the sum, and P (0) is 0.

Example 4. For the sample space { 1, 2, 3, 4, 5, 61 that represents the outcomes of record-

ing the number of pips on the top face after rolling a fair die, define

1

p(l) = p(2) = p(3) = p(4) = p(5) = p(6) = 6

Determine the probability that the top face shows an even number of pips, and determine

the probability that the number of pips on the top face is greater than four.

Solution. The event that the top face shows an even number of pips is E 1 = {2, 4, 6}, and

its probability is P(EI) = p(2) + p(4) + p(6) = 1/2. The event that the top face shows

more than four pips is E 2 = {5, 6}, and its probability is P(E 2 ) = p(5) + p(6) = 1/3. U

Note that for any event E, we have 0 < P(E) < 1. Also, note that P(Q2) = 1, and that

for a singleton event E = {w}, we have P(E) = p(co). (Why?)

8.1.3 Frequency Interpretation of Probability

In the preceding discussion, we defined the probability of an event, including that of a sin-

gleton event. The probability was defined in terms of a probability density function that can

be chosen arbitrarily, subject only to the two conditions in Definition 4 in Section 8.1.2. The

usefulness in practice of computations based on the definitions depends on how well the

chosen probability density function models the situation of interest. When we can imagine

doing a probabilistic experiment over and over, we generally choose pc(w) to estimate the

proportion of times that we think outcome a) will occur. When we compute the probability

of an event E, we generally interpret it as an estimate of the proportion of times that the

event should occur. This is called the frequency interpretation of probability.

The Frequency Interpretation of Probability

The frequency interpretation of probability is to take the quantity P(E) as an es-

timate for the proportion of times that event E will occur when an experiment is

repeated over and over. The reasonableness of the estimate depends on how well the

probability density function estimates the frequencies of the outcomes.

8.1.4 Introductory Example Reconsidered

To illustrate the difference between mathematical requirements and personal choices, let

us reconsider the sample spaces Q21 and Q22 from the dice-throwing example in light of the

definitions of the preceding subsection. In the 36-element sample space

Q2 = {(1, 1), (1, 2) ... , (6, 6))

obtaining a sum of 3 is described by the event E = {(1, 2), (2, 1)}, which breaks the situ-

ation into its smallest subcases, showing exactly how the 3 can be obtained. However, for

the sample space

Q1 = {2, 3, .... 121