1. Copyright © 2010 Pearson Education, Inc. Slide 17 - 1

2. Copyright © 2010 Pearson Education, Inc. Slide 17 - 2 Solution: D

3. Copyright © 2010 Pearson Education, Inc. Chapter 8 Probability Models

4. Copyright © 2010 Pearson Education, Inc. Slide 17 - 4 The Binomial Model A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of Bernoulli trials. Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p). Binomial Setting there are two possible outcomes (success and failure). the probability of success, p, is constant. the trials are independent and fixed.

5. Copyright © 2010 Pearson Education, Inc. Slide 17 - 5 TI – Tips Binomial Probabilities 2 nd DISTR Binompdf(n,p,X) Switches example pg. 442 P(X≤1) = Binompdf(10,.1,0) + Binompdf(10,.1,1) P(X≤1) = Binomcdf(10,.1,1)

6. Copyright © 2010 Pearson Education, Inc. Slide 17 - 6 The Binomial Model (cont.) In n trials, there are ways to have k successes. Read n C k as “n choose k.” Note: n! = n  (n – 1)  …  2  1, and n! is read as “n factorial.”

7. Copyright © 2010 Pearson Education, Inc. Slide 17 - 7 The Binomial Model (cont.) Binomial probability model for Bernoulli trials: Binom(n,p) n = number of trials p = probability of success q = 1 – p = probability of failure X = # of successes in n trials P(X = x) = n C x p x q n–x

8. Copyright © 2010 Pearson Education, Inc. Slide 17 - 8 Example: Postini reports that 91% of email messages are spam. Suppose your inbox contains 25 messages. What are the mean and standard deviation of the number of real messages you should expect to find in your inbox? What’s the probability that you’ll find only 1 or 2 real message?

9. Copyright © 2010 Pearson Education, Inc. Slide 17 - 9 The Normal Model to the Rescue! When dealing with a large number of trials in a Binomial situation, making direct calculations of the probabilities becomes tedious (or outright impossible). Fortunately, the Normal model comes to the rescue…

10. Copyright © 2010 Pearson Education, Inc. Slide 17 - 10 The Normal Model to the Rescue (cont.) As long as the Success/Failure Condition holds, we can use the Normal model to approximate Binomial probabilities. Success/failure condition: A Binomial model is approximately Normal if we expect at least 10 successes and 10 failures: np ≥ 10 and nq ≥ 10

11. Copyright © 2010 Pearson Education, Inc. Slide 17 - 11 The Geometric Model A Geometric probability model tells us the probability for a random variable that counts the number of Bernoulli trials until the first success. Geometric models are completely specified by one parameter, p, the probability of success, and are denoted Geom(p). Same setting as binomial (previous)

12. Copyright © 2010 Pearson Education, Inc. Geometric Distributions You want to draw an ace from a deck of cards P(X=1) P(X=2) Slide 17 - 12

13. Copyright © 2010 Pearson Education, Inc. Slide 17 - 13 The Geometric Model (cont.) Geometric probability model for Bernoulli trials: Geom(p) p = probability of success q = 1 – p = probability of failure X = number of trials until the first success occurs P(X = x) = q x-1 p

14. Copyright © 2010 Pearson Education, Inc. Slide 17 - 14 TI-Tips Geometric Probabilities 2 nd DISTR Geometpdf(p,x) “pdf” stands for “probability density function”

15. Copyright © 2010 Pearson Education, Inc. Slide 17 - 15 TI-Tips Geometric Probabilities 2 nd DISTR Geometcdf(p,x) “cdf” stands for “cumulative density function”

16. Copyright © 2010 Pearson Education, Inc. Slide 17 - 16 Example: Postini is company specializing in communications security. The company monitors over 1 billion Internet messages per day and recently reported that 91% of e-mails are spam! Let’s assume that your e- mail is typical, 91% spam. We’ll also assume you aren’t using a spam filter, so every message gets dumped into your inbox. Since spam comes from many different sources, we’ll consider your messages to be independent. Overnight your inbox collects email. When you first check your e-mail in the morning, about how many spam e-mails should you expect to find before you find a real message? Find the probability that the 4 th message in your inbox is the first one that isn’t spam.

17. Copyright © 2010 Pearson Education, Inc. Slide 17 - 17 Sum it up! Geometric model When we’re interested in the number of Bernoulli trials until the next success. Binomial model When we’re interested in the number of successes in a certain number of Bernoulli trials. Normal model To approximate a Binomial model when we expect at least 10 successes and 10 failures.

18. Copyright © 2010 Pearson Education, Inc. Slide 17 - 18 Example: We want to know the probability that we find our first Tiger Woods picture in the fifth box of cereal. Tiger is in 20% of the boxes. Example: Find the probability of getting a Tiger Woods picture by the time we open the fourth box of cereal. Example: We want to know the probability of finding Tiger exactly twice among five boxes of cereal. Example: Suppose we have ten boxes of cereal and we want to find the probability of finding up to 4 pictures of Tiger.

19. Copyright © 2010 Pearson Education, Inc. Slide 17 - 19 Example: Postini reported that 91% of e-mail messages are spam. Recently, you installed a spam filter. You observe that over the past week it okayed only 151 of 1422 e-mails you received, classifying the resent as junk. Should you worry the filtering is too aggressive? What’s the probability that no more than 151 of 1422 e-mails is a real message?