Other Types of Distributions

Transcription

1 Other Types of Distributions Unit 9 Probability Distributions Warm Up! The chance that a U.S. police chief believes the death penalty significantly reduces the number of homicides is 1 in 4. If a random sample of 8 police chiefs is selected, find the probability that at least 1 believe that the death penalty significantly reduces the number of homicides. 3 minutes 1

2 Warm Up! The chance that a U.S. police chief believes the death penalty significantly reduces the number of homicides is 1 in 4. If a random sample of 8 police chiefs is selected, find the probability that at least 1 believe that the death penalty significantly reduces the number of homicides. Answer: 1 P none = P at least 1 8! 8 0! 0! = P none = = The Multinomial Distribution Recall that in order for an experiment to be binomial, two outcomes are required for each trial. But if each trial in an experiment has more than two outcomes, a distribution called the multinomial distribution must be used. For example, a survey might require the responses of approve, disapprove, or no opinion. a person may have a choice of one of five activities for Friday night, such as a movie, dinner, baseball game, play, or party. 2

3 The Multinomial Distribution First, identify your sample size: n. Second, identify your events and the number of times each event occurs: X 9, X :, X <, etc. Third, identify the probability of each event: For event 1, the probability of happening or being chosen is p 9. For event 2, the probability of happening or being chosen is p :. For event 3, the probability of happening or being chosen is p <. Etc Then, for 3 events n! X 1! 0 X 2! 0 X 3! 0 p X 1 X 1 0 p 2 X 2 0 p 3 3 Leisure Activities Example 1 Find your variables: n, the X s and the p s. In a large city, 50% of the people choose a movie, 30% choose dinner and a play, and 20% choose shopping as a leisure activity. If a sample of 5 people is randomly selected, find the probability that 3 are planning to go to a movie, 1 to a play, and 1 to a shopping mall. 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 3

4 Leisure Activities Example 1 Find your variables: n, the X s and the p s. In a large city, 50% of the people choose a movie, 30% choose dinner and a play, and 20% choose shopping as a leisure activity. If a sample of 5 people is randomly selected, find the probability that 3 are planning to go to a movie, 1 to a play, and 1 to a shopping mall. Leisure Activities Example 1 Now, solve! In a large city, 50% of the people choose a movie, 30% choose dinner and a play, and 20% choose shopping as a leisure activity. If a sample of 5 people is randomly selected, find the probability that 3 are planning to go to a movie, 1 to a play, and 1 to a shopping mall. 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 4

5 Leisure Activities Example 1 Now, solve! In a large city, 50% of the people choose a movie, 30% choose dinner and a play, and 20% choose shopping as a leisure activity. If a sample of 5 people is randomly selected, find the probability that 3 are planning to go to a movie, 1 to a play, and 1 to a shopping mall. Nike purchase Example 2 Find your variables and solve! In a Nike store, a manager found that the probabilities that a person buys 0, 1, or 2 or more pair of shoes are 0.3, 0.6, and 0.1, respectively. If 6 customers enter the store, find the probability that 1 won t buy any shoes, 3 will buy 1 pair, and 2 will buy 2 or more pairs. 3 minutes 5

6 Nike purchase Example 2 Find your variables and solve! In a Nike store, a manager found that the probabilities that a person buys 0, 1, or 2 or more pair of shoes are 0.3, 0.6, and 0.1, respectively. If 6 customers enter the store, find the probability that 1 won t buy any shoes, 3 will buy 1 pair, and 2 will buy 2 or more pairs. Selecting Colored Balls Example 3 Find your variables and solve! A box contains 4 white balls, 3 red balls, and 3 blue balls. A ball is selected at random, and its color is written down. It is replaced each time. Find the probability that if 5 balls are selected, 2 are white, 2 are red, and 1 is blue. 3 minutes 6

7 Selecting Colored Balls Example 3 Find your variables and solve! A box contains 4 white balls, 3 red balls, and 3 blue balls. A ball is selected at random, and its color is written down. It is replaced each time. Find the probability that if 5 balls are selected, 2 are white, 2 are red, and 1 is blue. The Hypergeometric Distribution When sampling is done without replacement, the binomial distribution does not give exact probabilities, since the trials are not independent. The smaller the size of the population, the less accurate the binomial probabilities will be. The results of the problem can be generalized by using a special probability distribution called the hypergeometric distribution. The hypergeometric distribution is a distribution of a variable that has two outcomes when sampling is done without replacement. 7

8 The Hypergeometric Distribution Given a population with only two types of objects (females and males, defective and non-defective, successes and failures, etc.), such that there are a items of one kind and b items of another kind and a + b equals the total population, the probability P(X) of selecting without replacement a sample of size n with X items of type a and n X items of type b is Assistant Manager Applicants Example 4 Ten people apply for a job as assistant manager of a restaurant. Five have completed college and five have not. If the manager selects 3 applicants at random, find the probability that all 3 are college graduates. Setting up the variables: a = 5 college students n = 3 b = 5 nongraduates X = 3 n X = 0 5 minutes 8

9 Assistant Manager Applicants Example 4 Ten people apply for a job as assistant manager of a restaurant. Five have completed college and five have not. If the manager selects 3 applicants at random, find the probability that all 3 are college graduates. Setting up the variables: a = 5 college students n = 3 b = 5 nongraduates X = 3 n X = 0 House Insurance Example 5 A recent study found that 2 out of every 10 houses in a neighborhood have no insurance. If 5 houses are selected from 10 houses, find the probability that exactly 1 will be uninsured. 5 minutes 9

10 House Insurance Example 5 A recent study found that 2 out of every 10 houses in a neighborhood have no insurance. If 5 houses are selected from 10 houses, find the probability that exactly 1 will be uninsured. 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! 10