Robert P. Dobrow - Probability

Example 1.8 In a city, suppose 75% of the population have brown hair, 40% have brown eyes, and 25% have both brown hair and brown eyes. A person is chosen at random from the city. What is the probability that they

1 Have brown eyes or brown hair?

2 Have neither brown eyes nor brown hair?

To gain intuition, draw a Venn diagram, as in Figure 1.2. Let be the event of having brown hair; let denote brown eyes.

1 The probability of having brown eyes or brown hair isNotice that and are not mutually exclusive. If we made a mistake and used the simple addition rule , we would mistakenly get

2 The complement of having neither brown eyes nor brown hair is having brown eyes or brown hair. Thus,

FIGURE 1.2 : Venn diagram.

The simplest probability model for a finite sample space is that all outcomes are equally likely. If has elements, then the probability of each outcome is , as probabilities sum to 1. That is, , for all

Computing probabilities for equally likely outcomes takes a fairly simple form. Suppose is an event with elements, with . As is the sum of the probabilities of all the outcomes contained in ,

In other words, probability with equally likely outcomes reduces to counting elements in and .

Example 1.9 A palindrome is a word that reads the same forward or backward. Examples include mom, civic, and rotator. Pick a three-letter “word” at random choosing from D, O, or G for each letter. What is the probability that the resulting word is a palindrome? (Words in this context do not need to be real words in English, e.g., is a palindrome.)There are 27 possible words (three possibilities for each of the three letters). List and count the palindromes: DDD, OOO, GGG, DOD, DGD, ODO, OGO, GDG, and GOG. The probability of getting a palindrome is

Example 1.10 A bowl has red balls and blue balls. A ball is drawn randomly from the bowl. What is the probability of selecting a red ball?The sample space consists of balls. The event has elements. Therefore, .

A model for equally likely outcomes assumes a finite sample space. Interestingly, it is impossible to have a probability model of equally likely outcomes on an infinite sample space. To see why, suppose and for all , where is a nonzero constant. Then summing the probabilities gives

While equally likely outcomes are not possible in the infinite case, there are many ways to assign probabilities for an infinite sample space where outcomes are not equally likely. For instance, let with for . Then, using results for geometric series,

We introduce some basic counting principles in the next two sections because counting plays a fundamental role in probability when outcomes are equally likely.

Counting sets is sometimes not as easy as . But a basic counting principle known as the multiplication principle allows for tackling a wide range of problems.

If there are ways for one thing to happen, and ways for a second thing to happen, there are ways for both things to happen.

More generally—and more formally—consider an -element sequence If there are possible values for the first element, possible values for the second element, , and possible values for the th element, there are possible sequences.