Mean, Variance and Expected Value. Unit 9 Probability Distributions

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1 Mean, Variance and Expected Value Unit 9 Probability Distributions

2 5 minutes Warm Up! 1. A box contains 10 balls. Four are numbered 3, one is numbered 5, and five is numbered 4. The balls are mixed and one is drawn at random, its number recorded, then replaced. Construct the probability distribution of the drawing each ball. 2. One thousand tickets ($50 each) are sold for a raffle to win a car valued at $28,000. Construct the probability distribution for each ticket purchased.

3 Finding the Mean: Baseball World Series The following probability distribution represents the data for the number of games played in the MLB World Series from 1965 through (There was no World Series in 1994.) Find the mean. number of games played, X P(X) To find the mean of a probability distribution, add the products of the number and their corresponding probabilities: X. P(X). + X 0 P(X) 0 + X 1 P(X) 1 + X 2 P(X) 2 + +X 4 P(X) 4

4 Finding the Mean: Baseball World Series The following probability distribution represents the data for the number of games played in the MLB World Series from 1965 through (There was no World Series in 1994.) Find the mean. number of games played, X P(X) ANSWER: About 6 games X. P(X). + X 0 P(X) 0 + X 1 P(X) 1 + X 2 P(X) = 5.825

5 Number of Trips of Five Nights or More The probability distribution shown represents the number of trips of five nights or more that American adults take per year. (That is, 6% do not take any trips lasting five night or more, 70% take one trip lasting five nights or more per year, etc.) Find the mean. 3 minutes

Number of Trips of Five Nights or More The probability distribution shown represents the number of trips of five nights or more that American adults take

6 Number of Trips of Five Nights or More The probability distribution shown represents the number of trips of five nights or more that American adults take per year. (That is, 6% do not take any trips lasting five night or more, 70% take one trip lasting five nights or more per year, etc.) Find the mean. ANSWER:

7 Number of Trips of Five Nights or More If three coins are tossed, find the mean of the number of heads that occur. 1:59 1:58 1:57 1:56 1:55 1:54 1:53 1:52 1:51 1:50 1:49 1:48 1:47 1:46 1:45 1:44 1:43 1:42 1:41 1:40 1:39 1:38 1:37 1:36 1:35 1:34 1:33 1:32 1:31 1:30 1:29 1:28 1:27 1:26 1:25 1:24 1:23 1:22 1:21 1:20 1:19 1:18 1:17 1:16 1:15 1:14 1:13 1:12 1:11 1:10 1:09 1:08 1:07 1:06 1:05 1:04 1:03 1:02 1:01 1:00 0:59 0:58 0:57 0:56 0:55 0:54 0:53 0:52 0:51 0:50 0:49 0:48 0:47 0:46 0:45 0:44 0:43 0:42 0:41 0:40 0:39 0:38 0:37 0:36 0:35 0:34 0:33 0:32 0:31 0:30 0:29 0:28 0:27 0:26 0:25 0:24 0:23 0:22 0:21 0:20 0:19 0:18 0:17 0:16 0:15 0:14 0:13 0:12 0:11 0:10 0:09 0:08 0:07 0:06 0:05 0:04 0:03 0:02 0:01 2:00 End

8 Number of Trips of Five Nights or More If three coins are tossed, find the mean of the number of heads that occur. ANSWER:

9 On Hold for Talk Show A talk radio station has four telephone lines. If the host is unable to talk (i.e., during a commercial) or is talking to a person, the other callers are placed on hold. When all lines are in use, others who are trying to call in get a busy signal. The probability that 0, 1, 2, 3, or 4 people will get through is shown in the distribution. The standard deviation is 1.1. Find the mean and determine if the station should have considered getting more phone lines installed.

10 On Hold for Talk Show A talk radio station has four telephone lines. If the host is unable to talk (i.e., during a commercial) or is talking to a person, the other callers are placed on hold. When all lines are in use, others who are trying to call in get a busy signal. The probability that 0, 1, 2, 3, or 4 people will get through is shown in the distribution. The standard deviation is 1.1. Find the mean and determine if the station should have considered getting more phone lines installed. ANSWER: No. The mean number of people calling at any one time is 1.6. Since the standard deviation is 1.1, most callers would be accommodated by having four phone lines, because even with 2 standard deviations we have (1.1) = 3.8.

11 Expected Value EXAMPLE: One thousand tickets are sold at $1 each for a color television valued at $350. What is the expected value of the gain if you purchase one ticket?

12 Expected Value EXAMPLE: One thousand tickets are sold at $1 each for a color television valued at $350. What is the expected value of the gain if you purchase one ticket?

13 Winning Tickets One thousand tickets are sold at $1 each for four prizes of $100, $50, $25, and $10. After each prize drawing, the winning ticket is then returned to the pool of tickets. What is the expected value if you purchase two tickets? 3 minutes

14 Winning Tickets One thousand tickets are sold at $1 each for four prizes of $100, $50, $25, and $10. After each prize drawing, the winning ticket is then returned to the pool of tickets. What is the expected value if you purchase two tickets? ANSWER:

15 Bond Investment A financial adviser suggests that his client select one of two types of bonds in which to invest $5000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a 2.5% return and a default rate of 1%. Find the expected rate of return and decide which bond would be a better investment. When the bond defaults, the investor loses all the investment. 1:59 1:58 1:57 1:56 1:55 1:54 1:53 1:52 1:51 1:50 1:49 1:48 1:47 1:46 1:45 1:44 1:43 1:42 1:41 1:40 1:39 1:38 1:37 1:36 1:35 1:34 1:33 1:32 1:31 1:30 1:29 1:28 1:27 1:26 1:25 1:24 1:23 1:22 1:21 1:20 1:19 1:18 1:17 1:16 1:15 1:14 1:13 1:12 1:11 1:10 1:09 1:08 1:07 1:06 1:05 1:04 1:03 1:02 1:01 1:00 0:59 0:58 0:57 0:56 0:55 0:54 0:53 0:52 0:51 0:50 0:49 0:48 0:47 0:46 0:45 0:44 0:43 0:42 0:41 0:40 0:39 0:38 0:37 0:36 0:35 0:34 0:33 0:32 0:31 0:30 0:29 0:28 0:27 0:26 0:25 0:24 0:23 0:22 0:21 0:20 0:19 0:18 0:17 0:16 0:15 0:14 0:13 0:12 0:11 0:10 0:09 0:08 0:07 0:06 0:05 0:04 0:03 0:02 0:01 2:00 End

16 Bond Investment A financial adviser suggests that his client select one of two types of bonds in which to invest $5000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a 2.5% return and a default rate of 1%. Find the expected rate of return and decide which bond would be a better investment. When the bond defaults, the investor loses all the investment. ANSWER:

17 Partner/Independent Work Work on your worksheet. Try the exercises on your own. Help yourself be an independent learner/worker. You won t have your friends doing the exam for you! If you are struggling, then ask your partners for clues or true help. If you are still struggling, please ask me instead of just copying the answers!

Binomial Distributions

Binomial Distributions Unit 9 Probability Distributions 5 minutes Warm Up! 1. Read the Gambler s Fallacy: Why We Expect to Strike It Rich After a Losing Streak. 2. This study shows that a part of the brain