Discrete Mathematics for Computer Science

(^482) CHAPTER 8 Discrete Probability

How to Calculate the Probability of Events

1. Describe in words the experiment and the event or events of interest.


2. Choose a sample space Q2 so that the events of interest are easy to describe in Q.


3. Define a probability density function p on Q2.


4. Formulate the events of interest as subsets of ý2, and calculate their probabilities
   by summing the probability density function on the outcomes of the events.

8.1.5 The Combinatorics of Uniform Probability Density

Choosing a sample space with outcomes that do not decompose into more detailed subcases

makes it easy to express situations of interest as events in the sample space. Choosing such

a sample space has the further advantage that it is often easy to define an appropriate

probability density function on it. Whenever we have no reason to believe that one basic

situation is any more or less likely to occur than any other, we set p(w) = 1/1 Q I for each

) e Q provided that I[2I is finite. (Recall that putting vertical lines on each side of the

symbol for a finite set denotes the number of elements in that set.)

Definition 6. A probability density function p such that p(w) = 1/1 Q I for all a) in a

finite sample space Q2 is called a uniform probability density function.

For the standard deck of cards, it is useful to think of the cards as being represented by

the values 1, 2, 3 ... , 51, 52. Usually, we do not need to be concerned about which card

is represented by which value. We know, for example, that 26 of these values represent red

cards (hearts and diamonds) and that four of these values represent each card value. We

also assume that the deck is fair-that is, no card is more likely than any other to be chosen

in a random pick.

Example 5. Define a uniform probability density function on the standard deck of cards.

Determine the probability of drawing one of the 3's and the probability of drawing a face

card (Jack, Queen, or King).

Solution. For each card in the deck, p(card) = 1/52. The event El (that the card is a 3)

consists of four elements-3 of Clubs, 3 of Diamonds, 3 of Hearts, and 3 of Spades-so

P(Ei) = 1/13. The event E 2 (that the card is a face card) is a set consisting of 12 cards,

so P(E 2 ) = 12/52 = 3/13. 0

When a finite sample space ý2 is assigned a uniform probability density p, the combi-

natorial counting methods of Chapter 7 can provide shortcuts to computing the probability

of an event E. Since

P (E) = .,p(0) [ElE

o)EE

when Qi2 is finite and p is uniform, evaluating probabilities in this case is a matter of count-

ing set sizes.

The following examples illustrate the use of counting techniques to evaluate probabil-

ities. Throughout this chapter, unless stated otherwise, all coins, dice, and decks of cards