1. Chapter 12 Probability © 2008 Pearson Addison-Wesley. All rights reserved

2. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-2 Chapter 12: Probability 12.1 Basic Concepts 12.2 Events Involving “Not” and “Or” 12.3 Conditional Probability; Events Involving “And” 12.4 Binomial Probability 12.5Expected Value

3. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-3 Chapter 1 Section 12-4 Binomial Probability

4. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-4 Binomial Probability Binomial Probability Distribution Binomial Probability Formula

5. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-5 Binomial Probability Distribution The spinner below is spun twice and we are interested in the number of times a 2 is obtained (assume each sector is equally likely). Think of outcome 2 as a “success” and outcomes 1 and 3 as “failures.” The sample space is 1 2 3 S = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}.

6. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-6 Binomial Probability Distribution When the outcomes of an experiment are divided into just two categories, success and failure, the associated probabilities are called “binomial.” Repeated trials of the experiment, where the probability of success remains constant throughout all repetitions, are also known as Bernoulli trials.

7. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-7 Binomial Probability Distribution If x denotes the number of 2s occurring on each pair of spins, then x is an example of a random variable. In S, the number of 2s is 0 in four cases, 1 in four cases, and 2 in one case. Because the table on the next slide includes all the possible values of x and their probabilities it is an example of a probability distribution. In this case, it is a binomial probability distribution.

8. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-8 Probability Distribution for the Number of 2s in Two Spins xP(x)P(x) 0 1 2

9. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-9 Binomial Probability Formula In general, let n = the number of repeated trials, p = the probability of success on any given trial, q = 1 – p = the probability of failure on any given trial, andx = the number of successes that occur. Note that p remains fixed throughout all n trials. This means that all trials are independent. In general, x, successes can be assigned among n repeated trials in n C x different ways.

10. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-10 Binomial Probability Formula When n independent repeated trials occur, where p = probability of success and q = probability of failure with p and q (where q = 1 – p) remaining constant throughout all n trials, the probability of exactly x successes is given by

11. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-11 Example: Coin Tossing Find the probability of obtaining exactly three heads in five tosses of a fair coin. Solution This is a binomial experiment with n = 5, p = 1/2, q = 1/2, and x = 3.

12. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-12 Example: Rolling a Die Solution Find the probability of obtaining exactly two 3’s in six rolls of a fair die. This is a binomial experiment with n = 6, p = 1/6, q = 5/6, and x = 2.

13. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-13 Example: Rolling a Die Solution Find the probability of obtaining less than two 3’s in six rolls of a fair die. We have n = 6, p = 1/6, q = 5/6, and x < 2.

14. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-14 Example: Baseball Hits Solution A baseball player has a well-established batting average of.250. In the next series he will bat 10 times. Find the probability that he will get more than two hits. In this case n = 10, p =.250, q =.750, and x > 2.

15. © 2008 Pearson Addison-Wesley. All rights reserved 12-4-15 Example: Baseball Hits Solution (continued)