1. 1-1 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 1 Chapter 16 Probability Models

2. 1-2 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 2 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.

3. 1-3 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 3 Teaching Tip The term Bernoulli is not officially an AP topic. However, it is a good way to begin these topics. Don’t worry if students forget this name as you focus on the multiplication principle and binomial calculations.

4. 1-4 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 4 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).

5. 1-5 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 5 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

6. 1-6 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 6 Teaching tip Geometric probabilities are simply an extension of the multiplication principle. Do not focus on geometric formula, but rather on what students already know from the previous chapter. In fact, students already performed geometric probability calculations in Chapter 13 (#39 and following). You might want to start class with students redoing a couple of those problems, identifying them as geometric, and then diving into the binomial model as something they don’t know how to solve yet. For example, transition from “What are the chances he misses 3 free throws until he finally makes one? to “What are the chances he makes 7 out of 10?”.

7. 1-7 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 7 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.

8. 1-8 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 8 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).

9. 1-9 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 9 The Binomial Model (cont.) In n trials, there are ways to have k successes. Read n C k as “n choose k.” Note:, and n! is read as “n factorial.”

10. 1-10 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 10 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

11. 1-11 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 11 Teaching Tips The binomial formulas are on the official AP formula sheet. Which is provided for the entire AP exam. Give your students a copy! Note: The normal approximation of the binomial distribution is an optional AP topic. It is covered (differently) through sampling distributions of proportions in Chapter 17and you may want to skip it now and circle back and connect to it when you are in chapter 17.

12. 1-12 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 12 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…

13. 1-13 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 13 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

14. 1-14 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 14 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.

15. 1-15 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 15 What Can Go Wrong? Be sure you have Bernoulli trials. You need two outcomes per trial, a constant probability of success, and independence. Remember that the 10% Condition provides a reasonable substitute for independence. Don’t confuse Geometric and Binomial models. Don’t use the Normal approximation with small n. You need at least 10 successes and 10 failures to use the Normal approximation.

16. 1-16 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 16 What have we learned? Bernoulli trials show up in lots of places. Depending on the random variable of interest, we might be dealing with a Geometric model Binomial model Normal model

17. 1-17 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 17 What have we learned? (cont.) 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. 1-18 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 18 AP Tips The AP rubrics usually have three requirements for probability problems: Name of distribution Identification of correct parameters Correct calculation

19. 1-19 Copyright © 2015, 2010, 2007 Pearson Education, Inc. Chapter 16, Slide 19 AP Tips The Name can be identified with words (binomial) or with the proper formula (P(X = x) = n C x p x q n–x ) (Your teacher will probably ask you to write both and that’s a good thing!) The parameters should use standard notation: n = 10, p = 0.7 or μ = 65”, σ = 3” (for a normal problem)