Chapter. Discrete Probability Distributions Pearson Pren-ce Hall. All rights reserved

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1 Chapter 36 Discrete Probability Distributions 2010 Pearson Pren-ce Hall. All rights

2 Sec-on 6.1 Probability Rules 2010 Pearson Pren-ce Hall. All rights 6-2

3 2010 Pearson Pren-ce Hall. All rights 6-3

4 A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using lefers such as X Pearson Pren-ce Hall. All rights 6-4

5 A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plofed on a number line with space between each point. See the figure Pearson Pren-ce Hall. All rights 6-5

6 A con0nuous random variable has infinitely many values. The values of a con-nuous random variable can be plofed on a line in an uninterrupted fashion. See the figure Pearson Pren-ce Hall. All rights 6-6

7 EXAMPLE Distinguishing Between Discrete and Continuous Random Variables Determine whether the following random variables are discrete or con-nuous. State possible values for the random variable. (a) The number of light bulbs that burn out in a room of 10 light bulbs in the next year. Discrete; x = 0, 1, 2,, 10 (b) The number of leaves on a randomly selected Oak tree. Discrete; x = 0, 1, 2, (c) The length of -me between calls to 911. Con-nuous; t > Pearson Pren-ce Hall. All rights 6-7

10 2010 Pearson Pren-ce Hall. All rights 6-10

11 A probability distribu0on provides the possible values of the random variable X and their corresponding probabili-es. A probability distribu-on can be in the form of a table, graph or mathema-cal formula Pearson Pren-ce Hall. All rights 6-11

12 EXAMPLE A Discrete Probability Distribu-on The table to the right shows the probability distribution for the random variable X, where X represents the number of DVDs a person rents from a video store during a single visit. x P(x) Pearson Pren-ce Hall. All rights 6-12

13 2010 Pearson Pren-ce Hall. All rights 6-13

14 EXAMPLE Iden)fying Probability Distribu)ons Is the following a probability distribu-on? x P(x) Pearson Pren-ce Hall. All rights 6-14

15 EXAMPLE Iden)fying Probability Distribu)ons Is the following a probability distribu-on? x P(x) Pearson Pren-ce Hall. All rights 6-15

16 EXAMPLE Iden)fying Probability Distribu)ons Is the following a probability distribu-on? x P(x) Pearson Pren-ce Hall. All rights 6-16

19 2010 Pearson Pren-ce Hall. All rights 6-19

20 A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the ver-cal axis represents the probability of that value of the random variable Pearson Pren-ce Hall. All rights 6-20

21 EXAMPLE Drawing a Probability Histogram Draw a probability histogram of the probability distribu-on to the right, which represents the number of DVDs a person rents from a video store during a single visit. x P(x) Pearson Pren-ce Hall. All rights 6-21

22 2010 Pearson Pren-ce Hall. All rights 6-22

23 EXAMPLE Compu-ng the Mean of a Discrete Random Variable Compute the mean of the probability distribu-on to the right, which represents the number of DVDs a person rents from a video store during a single visit. x P(x) Pearson Pren-ce Hall. All rights 6-23

24 2010 Pearson Pren-ce Hall. All rights 6-24

25 The following data represent the number of DVDs rented by 100 randomly selected customers in a single visit. Compute the mean number of DVDs rented Pearson Pren-ce Hall. All rights 6-25

26 2010 Pearson Pren-ce Hall. All rights 6-26

27 As the number of trials of the experiment increases, the mean number of rentals approaches the mean of the probability distribu-on Pearson Pren-ce Hall. All rights 6-27

30 2010 Pearson Pren-ce Hall. All rights 6-30

31 Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the expected value, E(X), of the random variable Pearson Pren-ce Hall. All rights 6-31

32 EXAMPLE Compu-ng the Expected Value of a Discrete Random Variable A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a life insurance company sells a $250,000 one year term life insurance policy to a 49- year- old female for $530. According to the Na-onal Vital Sta-s-cs Report, Vol. 47, No. 28, the probability the female will survive the year is Compute the expected value of this policy to the insurance company. Survives Does not survive x P(x) ,000 = - 249, E(X) = 530( ) + (-249,470)( ) = $ Pearson Pren-ce Hall. All rights 6-32

33 Sec-on 6.2 The Binomial Probability Distribu-on 2010 Pearson Pren-ce Hall. All rights 6-33

34 2010 Pearson Pren-ce Hall. All rights 6-34

35 2010 Pearson Pren-ce Hall. All rights 6-35

36 EXAMPLE Iden)fying Binomial Experiments Which of the following are binomial experiments? (a) A player rolls a pair of fair die 10 -mes. The number X of 7 s rolled is recorded. Binomial experiment (b) The 11 largest airlines had an on- -me percentage of 84.7% in November, 2001 according to the Air Travel Consumer Report. In order to assess reasons for delays, an official with the FAA randomly selects flights un-l she finds 10 that were not on -me. The number of flights X that need to be selected is recorded. Not a binomial experiment not a fixed number of trials. (c) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded. Not a binomial experiment the trials are not independent Pearson Pren-ce Hall. All rights 6-36

37 Which of the following probability experiments represents a binomial experiment? A. Asking 50 adults the amount of their last cell phone bill B. Asking 10 prisoners the number of crimes for which they were convicted C. Asking 20 students the name of their favorite television show D. Asking 30 homeowners if they would favor a new tax to support education Slide 6-37 Copyright 2010 Pearson Educa-on, Inc.

38 Which of the following probability experiments represents a binomial experiment? A. Asking 50 adults the amount of their last cell phone bill B. Asking 10 prisoners the number of crimes for which they were convicted C. Asking 20 students the name of their favorite television show D. Asking 30 homeowners if they would favor a new tax to support education Slide 6-38 Copyright 2010 Pearson Educa-on, Inc.

39 2010 Pearson Pren-ce Hall. All rights 6-39

40 EXAMPLE Construc-ng a Binomial Probability Distribu-on According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time percentage of 79.0% in May, Suppose that 4 flights are randomly selected from May, 2008 and the number of on-time flights X is recorded. Construct a probability distribution for the random variable X using a tree diagram Pearson Pren-ce Hall. All rights 6-40

41 2010 Pearson Pren-ce Hall. All rights 6-41

42 2010 Pearson Pren-ce Hall. All rights 6-42

43 EXAMPLE Using the Binomial Probability Distribu-on Func-on According to the Experian Automotive, 35% of all car-owning households have three or more cars. (a) In a random sample of 20 car-owning households, what is the probability that exactly 5 have three or more cars? (b) In a random sample of 20 car-owning households, what is the probability that less than 4 have three or more cars? (c) In a random sample of 20 car-owning households, what is the probability that at least 4 have three or more cars? 2010 Pearson Pren-ce Hall. All rights 6-43

44 Forty-three percent of marriages end in divorce. You randomly select 15 married couples. Find the probability exactly 5 of the marriages will end in divorce. A B C D Slide 6-44 Copyright 2010 Pearson Educa-on, Inc.

45 Forty-three percent of marriages end in divorce. You randomly select 15 married couples. Find the probability exactly 5 of the marriages will end in divorce. A B C D Slide 6-45 Copyright 2010 Pearson Educa-on, Inc.

46 2010 Pearson Pren-ce Hall. All rights 6-46

47 2010 Pearson Pren-ce Hall. All rights 6-47

48 EXAMPLE Finding the Mean and Standard Devia)on of a Binomial Random Variable According to the Experian Automo-ve, 35% of all car- owning households have three or more cars. In a simple random sample of 400 car- owning households, determine the mean and standard devia-on number of car- owning households that will have three or more cars Pearson Pren-ce Hall. All rights 6-48

49 Forty-three percent of marriages end in divorce. You randomly select 15 married couples. Find the mean number of marriages that will end in divorce. A B C D Slide 6-49 Copyright 2010 Pearson Educa-on, Inc.

50 Forty-three percent of marriages end in divorce. You randomly select 15 married couples. Find the mean number of marriages that will end in divorce. A B C D Slide 6-50 Copyright 2010 Pearson Educa-on, Inc.

51 The mean number of customers arriving at a bank during a 15-minute period is 10. Find the probability that exactly 8 customers will arrive at the bank during a 15-minute period. A B C D Slide 6-51 Copyright 2010 Pearson Educa-on, Inc.

52 The mean number of customers arriving at a bank during a 15-minute period is 10. Find the probability that exactly 8 customers will arrive at the bank during a 15-minute period. A B C D Slide 6-52 Copyright 2010 Pearson Educa-on, Inc.

53 For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribu-on of the random variable X becomes bell- shaped. As a general rule of thumb, if np(1 p) > 10, then the probability distribu-on will be approximately bell- shaped Pearson Pren-ce Hall. All rights 6-53

54 2010 Pearson Pren-ce Hall. All rights 6-54

EXAMPLE Using the Mean, Standard Devia)on and Empirical Rule to Check for Unusual Results in a Binomial Experiment According to the Experian Automotive, 35% of all car-owning households have three or

55 EXAMPLE Using the Mean, Standard Devia)on and Empirical Rule to Check for Unusual Results in a Binomial Experiment According to the Experian Automotive, 35% of all car-owning households have three or more cars. A researcher believes this percentage is higher than the percentage reported by Experian Automotive. He conducts a simple random sample of 400 car-owning households and found that 162 had three or more cars. Is this result unusual? The result is unusual since 162 > Pearson Pren-ce Hall. All rights 6-55

A random variable is a (typically represented by ) that has a. value, determined by, A probability distribution is a that gives the

5.2 RANDOM VARIABLES A random variable is a (typically represented by ) that has a value, determined by, for each of a. A probability distribution is a that gives the for each value of the. It is often