[Stefan Waner, Steven Costenoble] Finite …

7

Probability

7.1 Sample Spaces and Events

7.2 Relative Frequency 7.3 Probability and

Probability Models

7.4 Probability and Counting Techniques

7.5 Conditional Probability and Independence

7.6 Bayes' Theorem and Applications

7.7 Markov Systems

KEY CONCEPTS REVIEW EXERCISES CASE STUDY TECHNOLOGY GUIDES

Case Study The Monty Hall Problem

On the game show Let's Make a Deal, you are shown three doors, A, B, and C, and behind one of them is the Big Prize. After you select one of them--say, door A--to make things more interesting the host (Monty Hall) opens one of the other doors--say, door B--revealing that the Big Prize is not there. He then offers you the opportunity to change your selection to the remaining door, door C. Should you switch or stick with your original guess? Does it make any difference?

Everett Collection

Web Site At the Web site you will find:

? Section by section tutorials, including game tutorials with randomized quizzes

? A detailed chapter summary

? A true/false quiz

? A Markov system simulation and matrix algebra tool

? Additional review exercises 445

446 Chapter 7 Probability

Introduction

What is the probability of winning the lottery twice? What are the chances that a college athlete whose drug test is positive for steroid use is actually using steroids? You are playing poker and have been dealt two iacks. What is the likelihood that one of the next three cards you are dealt will also be a jack? These are all questions about probability.

Understanding probability is important in many fields, ranging from risk management in business through hypothesis testing in psychology to quantum mechanics in physics. Historically, the theory of probability arose in the sixteenth and seventeenth centuries from attempts by mathematicians such as Gerolamo Cardano, Pierre de Fermat, Blaise Pascal, and Christiaan Huygens to understand games of chance. Andrey Nikolaevich Kolmogorov set forth the foundations of modern probability theory in his 1933 book Foundations of the Theory of Probability.

The goal of this chapter is to familiarize you with the basic concepts of modern probability theory and to give you a working knowledge that you can apply in a variety of situations. In the first two sections, the emphasis is on translating real-life situations into the language of sample spaces, events, and probability. Once we have mastered the language of probability, we spend the rest of the chapter studying some of its theory and applications. The last section gives an interesting application of both probability and matrix arithmetic.

7.1 Sample Spaces and Events

Sample Spaces

At the beginning of a football game, to ensure fairness, the referee tosses a coin to decide who will get the ball first. When the ref tosses the coin and observes which side faces up, there are two possible results: heads (H) and tails (T). These are the only possible results, ignoring the (remote) possibility that the coin lands on its edge. The act of tossing the coin is an example of an experiment. The two possible results, H and T, are possible outcomes of the experiment, and the set S = {H, T} of all possible outcomes is the sample space for the experiment.

Experiments, Outcomes, and Sample Spaces

An experiment is an occurrence with a result, or outcome, that is uncertain before the experiment takes place. The set of all possible outcomes is called the sample space for the experiment.

Quick Examples

1. Experiment: Flip a coin and observe the side facing up. Outcomes: H, T Sample Space: S = {H, T}

2. Experiment: Select a student in your class. Outcomes: The students in your class Sample Space: The set of students in your class

7.1 Sample Spaces and Events 447

3. Experiment: Select a student in your class and observe the color of his or

her hair. Outcomes: red, black, brown, blond, green, . . . Sample Space: {red, black, brown, blond, green, . . .}

4. Experiment: Cast a die and observe the number facing up. Outcomes: 1, 2, 3, 4, 5, 6 Sample Space: S = {1, 2, 3, 4, 5, 6}

5. Experiment: Cast two distinguishable dice (see Example 1(a) of Sec-

tion 6.1) and observe the numbers facing up.

Outcomes: (1, 1), (1, 2), . . . , (6, 6) (36 outcomes)

(1, (2, (3,

1), 1), 1),

(1, 2), (2, 2), (3, 2),

(1, 3), (2, 3), (3, 3),

(1, 4), (2, 4), (3, 4),

(1, 5), (2, 5), (3, 5),

(1, (2, (3,

6), 6), 6),

Sample

Space:

S

=

(4, (5, (6,

1), 1), 1),

(4, 2), (5, 2), (6, 2),

(4, 3), (5, 3), (6, 3),

(4, 4), (5, 4), (6, 4),

(4, 5), (5, 5), (6, 5),

(4, (5, (6,

6), 6), 6)

n(S) = the number of outcomes in S = 36

6. Experiment: Cast two indistinguishable dice (see Example 1(b) of Sec-

tion 6.1) and observe the numbers facing up.

Outcomes: (1, 1), (1, 2), . . . , (6, 6) (21 outcomes)

Sample Space:

(1, 1), S =

(1, 2), (2, 2),

(1, 3), (2, 3), (3, 3),

(1, 4), (2, 4), (3, 4), (4, 4),

(1, 5), (2, 5), (3, 5), (4, 5), (5, 5),

(1, (2, (3,

6), 6), 6),

(4, (5, (6,

6), 6), 6)

n(S) = 21

7. Experiment: Cast two dice and observe the sum of the numbers facing up. Outcomes: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Sample Space: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

8. Experiment: Choose 2 cars (without regard to order) at random from a fleet of 10. Outcomes: Collections of 2 cars chosen from 10 Sample Space: The set of all collections of 2 cars chosen from 10

n(S) = C(10, 2) = 45

The following example introduces a sample space that we'll use in several other examples.

Cindy Charles/PhotoEdit

448 Chapter 7 Probability

EXAMPLE 1 School and Work

In a survey conducted by the Bureau of Labor Statistics,* the high school graduating class of 2007 was divided into those who went on to college and those who did not. Those who went on to college were further divided into those who went to two-year colleges and those who went to four-year colleges. All graduates were also asked whether they were working or not. Find the sample space for the experiment "Select a member of the high school graduating class of 2007 and classify his or her subsequent school and work activity."

Solution The tree in Figure 1 shows the various possibilities.

College

No College

Figure 1

2-year 4-year

Working Not Working

Working Not Working

Working Not Working

The sample space is

S = {2-year college & working, 2-year college & not working, 4-year college & working, 4-year college & not working, no college & working, no college & not working}.

*"College Enrollment and Work Activity of High School Graduates News Release," U.S. Bureau of Labor Statistics, April 2008, available at news.release/hsgec.htm.

Events

In Example 1, suppose we are interested in the event that a 2003 high school graduate was working. In mathematical language, we are interested in the subset of the sample space consisting of all outcomes in which the graduate was working.

Events

Given a sample space S, an event E is a subset of S. The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E--that is, if the outcome of the experiment is favorable.

7.1 Sample Spaces and Events 449

Visualizing an Event In the following figure, the favorable outcomes (events in E) are shown in green.

Sample Space S

E

Quick Examples 1. Experiment: Roll a die and observe the number facing up. S = {1, 2, 3, 4, 5, 6} Event: E: The number observed is odd. E = {1, 3, 5}

2. Experiment: Roll two distinguishable dice and observe the numbers facing up. S = {(1, 1), (1, 2), . . . , (6, 6)} Event: F: The dice show the same number. F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

3. Experiment: Roll two distinguishable dice and observe the numbers facing up.

S = {(1, 1), (1, 2), . . . , (6, 6)}

Event: G: The sum of the numbers is 1.

G=

There are no favorable outcomes.

4. Experiment: Select a city beginning with "J." Event: E: The city is Johannesburg.

E = {Johannesburg}

An event can consist of a single outcome.

5. Experiment: Roll a die and observe the number facing up. Event: E: The number observed is either even or odd.

E = S = {1, 2, 3, 4, 5, 6}

An event can consist of all possible outcomes.

6. Experiment: Select a student in your class. Event: E: The student has red hair.

E = {red-haired students in your class}

7. Experiment: Draw a hand of two cards from a deck of 52. Event: H: Both cards are diamonds.

H is the set of all hands of 2 cards chosen from 52 such that both cards are diamonds.

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