2. Copyright © 2009 Pearson Education, Inc. Chapter 17 Probability Models

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: The student will be able to: Tell if a situation involves Bernoulli trials. Know the appropriate conditions for using a Binomial or Normal model. Find and interpret in context the mean and standard deviation of a Binomial model. Calculate binomial probabilities, perhaps with a Normal model.

4. Copyright © 2009 Pearson Education, Inc. Slide 1- 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. Examples: Flipping a coin (where heads is success), rolling a die (where getting a “6” is success), throwing free throws in a basketball game, drawing a card from a deck of cards with replacement (where drawing an Ace is success)

5. Copyright © 2009 Pearson Education, Inc. Do we have Bernoulli Trials? You are rolling 5 dice and need to get at least two 6’s to win the game We record the eye colors found in a group of 500 people A city council of 11 Republicans and 8 Democrats picks a committee of 4 at random. What is the probability that they choose all Democrats? A 2002 Rutgers University study found that 74% of high school students have cheated on a test at least once. Your local high school principle conducts a survey and gets responses that admit to cheating from 322 of 481 students. How likely is it that in a group of 120 the majority may have type A blood, given that Type A is found in 43% of the population? Slide 1- 5

6. Copyright © 2009 Pearson Education, Inc. Slide 1- 6 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. Example: lets draw cards from a standard deck with replacement and consider drawing a heart “success.” Do we have Bernoulli trials? Would we have Bernoulli trials if we were drawing without replacement? What is the probability p of success? What is the probability q of failure? What is the probability that the first heart is the 3 rd card drawn? i.e. first success occurs on trial 3. Geometric models are completely specified by one parameter, p, the probability of success, and are denoted Geom(p).

7. Copyright © 2009 Pearson Education, Inc. Slide 1- 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 In our example P(X=3) = (39/52) 2 (13/52)

8. Copyright © 2009 Pearson Education, Inc. Slide 1- 8 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.

9. Copyright © 2009 Pearson Education, Inc. Slide 1- 9 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. Example: If success is drawing a heart (drawing with replacement), what is the probability that if we draw and replace 3 cards that we drew exactly one heart? Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p). Example: If we flip a coin 6 times what is the probability of getting heads exactly 3 times?

10. Copyright © 2009 Pearson Education, Inc. Slide 1- 10 The Binomial Model (cont.) In n trials, there are ways to have k successes. Read n C k as “n choose k,” and is called a combination. Example: How many ways are there to roll a die five times and roll a 6 three of those times? Note: n! = n x (n – 1) x … x 2 x 1, and n! is read as “n factorial.”

11. Copyright © 2009 Pearson Education, Inc. Slide 1- 11 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 = number of successes in n trials

12. Copyright © 2009 Pearson Education, Inc. Slide 1- 12 Using the TI Suppose a light bulb company has a 20% defective rate. Consider taking a sample of 6 bulbs. 1) What is the probability of getting exactly 1 defective bulb in that group of 6? (Even though a defect isn't pleasant at times, it is considered a success in this experiment since that is where our focus is!) If we computed this probability long hand we would do 6 C 1 (.20) 1 (.80) 5 On the calculator: 2nd Distr...(#0) for binompdf(6,.2, 1)... enter to get.393216 binompdf gives you the probability at a particular x. The pdf must be followed by n (total number of trials), p (probability of a success), x (number of successes you are interested in) binomcdf, which we use next, will compute cumulative probabilities.

13. Copyright © 2009 Pearson Education, Inc. Slide 1- 13 Using the TI 2) What is the probability of getting at most 2 defective light bulbs? This means P(0) + P(1) + P(2) OR use the cumulative binomial button: 2nd Distr...#A for binomcdf(6,.2, 2)...enter to get.90112 3) What is the probability of getting at least two defective light bulbs? At least two means two or more which is the same as adding the probabilities of 2 to 3 to 4 etc... OR 1 minus the complement of "at least two" which is 1 minus the cdf to 1 1 - binomcdf (6,.2,1) =.34464 4) What is the probability of getting from two to four defective light bulbs? You could do the pdf for 2 + pdf for 3 + pdf for 4 or be a little creative and do binmocdf(6,.2,4) - binomcdf(6,.2,1) =.34304

14. Copyright © 2009 Pearson Education, Inc. Using StatCrunch To use StatCrunch to calculate Binomial Probabilities (or to view the binomial probability histogram) go to Stat -> Calculators -> Binomial Enter the appropriate n, p, and Prob statement. Then click "Calculate" For example, if you want to compute the probability of observing at least one "6" in 5 rolls of the die, n = 5 p = 0.1667 Prob (X=>1) = 0.598 Slide 1- 14

15. Copyright © 2009 Pearson Education, Inc. 20) An Olympic archer is able to hit the bull’s-eye 80% of the time. Assume each shot is independent of the others. If she shoots 6 arrows, what’s the probability of the following Her first bull’s-eye comes on the 3 rd arrow (note: this uses the Geometric not the Binomial Distribution) She misses the bull’s-eye at least once Her first bull’s-eye comes on the fourth or fifth arrow (Geometric) She gets exactly 4 bull’s-eyes She gets at least 4 bull’s-eyes She gets at most 4 bull’s-eyes Slide 1- 15

16. Copyright © 2009 Pearson Education, Inc. 17) If you flip a fair coin 100 times Intuitively how many heads do you expect? Use the formula for expected value to verify your intuition 18) An American roulette wheel has 38 slots, of which 18 are red, 18 are black, and 2 are green. If you spin the wheel 38 times Intuitively how many times do you expect the ball to land in a green slot? Use the formula for expected value to verify your intuition Slide 1- 16

17. Copyright © 2009 Pearson Education, Inc. 22a, b Consider the same archer How many Bull’s-eyes do you expect her to get? With what standard deviation? Suppose our archer shoots 10 arros Find the mean and standard deviation of the number of bull’s-eyes you may get What’s the probability that she never misses? What the probability that there are no more than 8 bull’s-eyes What’s the probability that there are exactly 8 bull’s- eyes What’s the probability that she hits the bull’e-eye more often than she misses Slide 1- 17

18. Copyright © 2009 Pearson Education, Inc. Practice 6) Suppose 75% of all drivers always wear their seatbelts. Lets investigate how many of the drivers might be belted among six cars waiting at a traffic light. Describe how you’ll simulate the number of seatbelt wearing drivers among the six cars Run 30+ trials Based on the simulation estimate the probabilities that there are exactly no belted drivers, one, two, three, etc. Calculate the actual probability model Compare the distribution of outcomes in the simulation to the actual model Slide 1- 18

19. Copyright © 2009 Pearson Education, Inc. Recall our election example from chapter 11 If your candidate is favored by approximately 53% of the population, but only 100 people vote, what is the probability that your candidate wins? In other words, your candidate needs at least 51 votes of 100 votes. Assume each voter is independent. Do we have Bernoulli trials? What is the probability of “success” for a trial? What the probability that at least 51 of 100 voters vote for your candidate? Slide 1- 19

20. Copyright © 2009 Pearson Education, Inc. Slide 1- 20 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…

21. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 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.

22. Copyright © 2009 Pearson Education, Inc. Slide 1- 22 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. Ex. For our election example: μ = np = 100*.53 =53 σ = √(npq) = √(100*.53*.47) =4.99 P(at least 51 votes) ~ Normalcdf(51,100, 53, 4.99)

23. Copyright © 2009 Pearson Education, Inc. Slide 1- 23 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.

24. Copyright © 2009 Pearson Education, Inc. Slide 1- 24 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 Poisson model

25. Copyright © 2009 Pearson Education, Inc. Slide 1- 25 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. Poisson model To approximate a Binomial model when the probability of success, p, is very small and the number of trials, n, is very large.