1. Copyright © Cengage Learning. All rights reserved. The Binomial Probability Distribution and Related Topics 6

2. Copyright © Cengage Learning. All rights reserved. Section 6.2 Binomial Probabilities

3. 3 Focus Points List the defining features of a binomial experiment. Compute binomial probabilities using the formula P (r) = C n,r p r q n – r Use the binomial table to find P (r). Use the binomial probability distribution to solve real-world applications.

4. 4 Binomial Experiment

5. 5 On a TV game show, each contestant has a try at the wheel of fortune. The wheel of fortune is a roulette wheel with 36 slots, one of which is gold. If the ball lands in the gold slot, the contestant wins $50,000. No other slot pays. What is the probability that the game show will have to pay the fortune to three contestants out of 100? In this problem, the contestant and the game show sponsors are concerned about only two outcomes from the wheel of fortune: The ball lands on the gold, or the ball does not land on the gold.

6. 6 Binomial Experiment This problem is typical of an entire class of problems that are characterized by the feature that there are exactly two possible outcomes (for each trial) of interest. These problems are called binomial experiments, or Bernoulli experiments, after the Swiss mathematician Jacob Bernoulli, who studied them extensively in the late 1600s.

7. 7 Example 3 – Binomial Experiment Let’s see how the wheel of fortune problem meets the criteria of a binomial experiment. We’ll take the criteria one at a time. Solution: 1. Each of the 100 contestants has a trial at the wheel, so there are n = 100 trials in this problem. 2. Assuming that the wheel is fair, the trials are independent, since the result of one spin of the wheel has no effect on the results of other spins.

8. 8 Example 3 – Solution 3. We are interested in only two outcomes on each spin of the wheel: The ball either lands on the gold, or it does not. Let’s call landing on the gold success (S) and not landing on the gold failure (F). In general, the assignment of the terms success and failure to outcomes does not imply good or bad results. These terms are assigned simply for the user’s convenience. cont’d

9. 9 Example 3 – Solution 4. On each trial the probability p of success (landing on the gold) is 1/36, since there are 36 slots and only one of them is gold. Consequently, the probability of failure is on each trial. cont’d

10. 10 Example 3 – Solution 5. We want to know the probability of 3 successes out of 100 trials, so in this example. It turns out that the probability the quiz show will have to pay the fortune to r = 3 contestants out of 100 is about 0.23. Later in this section we’ll see how this probability was computed. cont’d

11. 11 Is it a binomial experiment? 1.An experiment in which a basketball player who historically makes 80% of his free throws is asked to shoot 3 free throws and the number of made free throws is recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____q = _____ n = _____x = _______________

12. 12 Is it a binomial experiment? 2. The number of people with blood type O-negative based upon a simple random sample of size 10 is recorded. According to the Information Please Almanac, 6% of the human population is blood type 0-negative. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____q = _____ n = _____x = _______________

13. 13 Is it a binomial experiment? 3. A probability experiment in which three cards are drawn from a deck without replacement and the number of aces is recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____q = _____ n = _____x = _______________

14. 14 Is it a binomial experiment? 4. A random sample of 15 college seniors is conducted, and the individuals selected are asked to state their ages. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____q = _____ n = _____x = _______________

15. 15 Is it a binomial experiment? 5. An experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____q = _____ n = _____x = _______________

16. 16 Is it a binomial experiment? 6.A poll of 1200 registered voters is conducted in which the respondents are asked whether they believe Congress shoulc reform Social Security. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____q = _____ n = _____x = _______________

17. 17 Is it a binomial experiment? 7. A baseball player who reaches base safely 30% of the time is allowed to bat until he reaches base safely for the third time. The number of at-bats required is recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____q = _____ n = _____x = _______________

18. 18 Is it a binomial experiment? 8. An investor randomly purchases 10 stocks listed on the New York Stock Exchange. Historically, the probability that a stock listed on the NYSE will increase in value over the course of a year is 48%. The number of stocks that increase in value is recorded. There are a fixed number of trials? Each trial is independent? Each trial has 2 outcomes? The probability of success for each trial is the same? If yes: p = _____q = _____ n = _____x = _______________

19. 19 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula

20. 20 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula The central problem of a binomial experiment is finding the probability of r successes out of n trials. Now we’ll see how to find these probabilities. Suppose you are taking a timed final exam. You have three multiple-choice questions left to do. Each question has four suggested answers, and only one of the answers is correct. You have only 5 seconds left to do these three questions, so you decide to mark answers on the answer sheet without even reading the questions.

21. 21 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Assuming that your answers are randomly selected, what is the probability that you get zero, one, two, or all three questions correct? This is a binomial experiment. Each question can be thought of as a trial, so there are n = 3 trials. The possible outcomes on each trial are success S, indicating a correct response, or failure F, meaning a wrong answer. The trials are independent—the outcome of any one trial does not affect the outcome of the others.

22. 22 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula What is the probability of success on anyone question? Since you are guessing and there are four answers from which to select, the probability of a correct answer is 0.25. The probability q of a wrong answer is then 0.75. In short, we have a binomial experiment with n = 3, p = 0.25, and q = 0.75. Now, what are the possible outcomes in terms of success or failure for these three trials? Let’s use the notation SSF to mean success on the first question, success on the second, and failure on the third.

23. 23 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula There are eight possible combinations of S’s and F’s. They are SSS SSF SFS FSS SFF FSF FFS FFF To compute the probability of each outcome, we can use the multiplication law because the trials are independent. For instance, the probability of success on the first two questions and failure on the last is P(SSF) = P(S)  P(S)  P(F) = p  p  q = p 2 q = (0.25) 2 (0.75)  0.047

24. 24 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula In a similar fashion, we can compute the probability of each of the eight outcomes. These are shown in Table 6-8, along with the number of successes r associated with each outcome. Outcomes for a Binomial Experiment with n = 3 Trials Table 6-8

25. 25 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Now we can compute the probability of r successes out of three trials for r = 0, 1, 2, or 3. Let’s compute P(1). The notation P(1) stands for the probability of one success. For three trials, there are three different outcomes that show exactly one success. They are the outcomes SFF, FSF, and FFS.

26. 26 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Since the outcomes are mutually exclusive, we can add the probabilities. So, P(1) = P(SFF or FSF or FFS) = P(SFF) + P(FSF) + P(FFS) = pq 2 + pq 2 + pq 2 = 3pq 2 = 3(0.25)(0.75) 2 = 0.422

27. 27 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula In the same way, we can find P(0), P(2), and P(3). These values are shown in Table 6-9. P(r) for n = 3 Trials, p = 0.25 Table 6-9

28. 28 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula

29. 29 Example 4 – Compute P(r) using the binomial distribution formula Privacy is a concern for many users of the Internet. One survey showed that 59% of Internet users are somewhat concerned about the confidentiality of their e-mail. Based on this information, what is the probability that for a random sample of 10 Internet users, 6 are concerned about the privacy of their e-mail? Solution: a. This is a binomial experiment with 10 trials. If we assign success to an Internet user being concerned about the privacy of e-mail, the probability of success is 59%. We are interested in the probability of 6 successes.

30. 30 Example 4 – Solution We have n = 10 p = 0.59 q = 0.41 r = 6 By the formula, P(6) = C 10,6 (0.59) 6 (0.41) 10 – 6 = 210(0.59) 6 (0.41) 4  210(0.0422)(0.0283)  0.25 There is a 25% chance that exactly 6 of the 10 Internet users are concerned about the privacy of e-mail. Use a calculator or the formula for c n,r. Use a calculator. cont’d

31. 31 Example 4 – Solution b. Many calculators have a built-in combinations function. On the TI-84Plus/TI-83Plus/TI-nspire (with TI-84Plus keypad) calculators, press the MATH key and select PRB. The combinations function is designated nCr. Figure 6-2 displays the process for computing P(6) directly on these calculators. TI-84Plus/TI-83Plus/TI-nspire (with TI-84Plus keypad) Display Figure 6-2 cont’d