Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 17- 1

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 17 Probability Models

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Bernoulli Trials An experiment consists of flipping a fair coin 8 times and counting the number of tails. Does this experiment consist of Bernoulli trials? A multiple choice test contains 20 questions. Each question has five choices for the correct answer. Only one of the choices is correct. With random guessing, does the test consist of Bernoulli trials?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Bernoulli Trials John buys packs of football cards. The probability that a pack contains a Donovan McNabb card is 3%. Does opening packs of cards and keeping track of McNabb cards consist of Bernoulli trials? A multiple choice test contains 20 questions. Each question has four or five choices for the correct answer. Only one of the choices is correct. With random guessing, does this test consist of Bernoulli trials?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Geometric Model A single Bernoulli trial is usually not all that interesting. 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).

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Geometric Model (cont.) Geometric probability model for Bernoulli trials: Geom(p) p = probability of success q = 1 – p = probability of failure X = # of trials until the first success occurs P(X = x) = q x-1 p

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Geometric Model (cont.) John buys 10 packs of football cards. The probability that a pack contains a Donovan McNabb card is 3%. Find the probability that John finds the first McNabb card in the 5 th pack; in the 10 th pack; not at all; in the first 6 packs; somewhere in packs 4 though 6.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Geometric Model (cont.) John plans to buy packs of football cards until he finds a Donovan McNabb card. The probability that a pack contains a Donovan McNabb card is 3%. What is the expected number of packs that John will purchase? What is the standard deviation of the number of packs that John will purchase? Would it be unusual if John purchased only 8 packs before finding a McNabb card?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Independence One of the important requirements for Bernoulli trials is that the trials be independent. When we don’t 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.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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).

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Binomial Model (cont.) An experiment consists of flipping a fair coin 8 times and counting the number of tails. Find the probability of seeing exactly 3 tails. Find the probability of seeing at most 3 tails. Find the probability of seeing at least 3 tails. Find the probability of seeing between 5 and 7 tails.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Binomial Model (cont.) A true/false choice test contains 20 questions. What is the probability of making an 80 with random guessing? What is the probability of making at least an 80 with random guessing? What is the probability of scoring between 60 and 70 (inclusive) with random guessing? Would it be unusual to score a 65 with random guessing?