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 17 Probability Models

4. Copyright © 2010 Pearson Education, Inc. Slide 17 - 4 Bernoulli Trials The basis for the probability models we will examine in this chapter is the Bernoulli trial. We have Bernoulli trials if: there are two possible outcomes (success and failure). the probability of success, p, is constant. the trials are independent.

5. Copyright © 2010 Pearson Education, Inc. Slide 17 - 5 Independence One of the important requirements for Bernoulli trials is that the trials be independent. When we dont have an infinite population, the trials are not independent. But, there is a rule that allows us to pretend we have independent trials: The 10% condition: Bernoulli trials must be independent. If that assumption is violated, it is still okay to proceed as long as the sample is smaller than 10% of the population.

6. Copyright © 2010 Pearson Education, Inc. Slide 17 - 6 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).

7. Copyright © 2010 Pearson Education, Inc. Slide 17 - 7 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

8. Copyright © 2010 Pearson Education, Inc. Slide 17 - 8 TI-Tips Geometric Probabilities 2 nd DISTR Geometpdf(p,x) pdf stands for probability density function This allows us to find probabilities such as The probability of finding our 1 st in the 5 th box.

9. Copyright © 2010 Pearson Education, Inc. Slide 17 - 9 TI-Tips Geometric Probabilities 2 nd DISTR Geometcdf(p,x) cdf stands for cumulative density function This allows us to find probabilities such as The probability of finding at least one by the 5 th box.

10. Copyright © 2010 Pearson Education, Inc. Slide 17 - 10 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! Lets assume that your e- mail is typical, 91% spam. Well also assume you arent using a spam filter, so every message gets dumped into your inbox. Since spam comes from many different sources, well 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 isnt spam.

11. Copyright © 2010 Pearson Education, Inc. Slide 17 - 11 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).

12. Copyright © 2010 Pearson Education, Inc. Slide 17 - 12 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.

13. Copyright © 2010 Pearson Education, Inc. Slide 17 - 13 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

14. Copyright © 2010 Pearson Education, Inc. Slide 17 - 14 TI – Tips Binomial Probabilities 2 nd DISTR Binompdf(n,p,X) This finds the probability of finding exactly 2 in 5 boxes

15. Copyright © 2010 Pearson Education, Inc. Slide 17 - 15 TI – Tips Binomial Probabilities 2 nd DISTR Binomcdf(n,p,X) This finds the probability of finding up to 4 in 5 boxes or no more than 4 in 5 boxes To find: At least 5 = 1-binomcdf(n,p,4)

16. Copyright © 2010 Pearson Education, Inc. Slide 17 - 16 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? Whats the probability that youll find only 1 or 2 real message?

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

18. Copyright © 2010 Pearson Education, Inc. Slide 17 - 18 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

19. Copyright © 2010 Pearson Education, Inc. Slide 17 - 19 Continuous Random Variables When we use the Normal model to approximate the Binomial model, we are using a continuous random variable to approximate a discrete random variable. So, when we use the Normal model, we no longer calculate the probability that the random variable equals a particular value, but only that it lies between two values.

20. Copyright © 2010 Pearson Education, Inc. Slide 17 - 20 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? Whats the probability that no more than 151 of 1422 e-mails is a real message?

21. Copyright © 2010 Pearson Education, Inc. Slide 17 - 21 Sum it up! Geometric model When were interested in the number of Bernoulli trials until the next success. Binomial model When were 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.

22. Copyright © 2010 Pearson Education, Inc. Slide 17 - 22 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.

23. Copyright © 2010 Pearson Education, Inc. Slide 17 - 23 Homework: Pg. 401 1-41 odd