chapter 36rlhawkmath.pbworks.com/w/file/fetch/65341448/ch6 rl.pdf · 2020. 11. 29. · chapter...

Chapter

Discrete Probability Distributions

© 2010 Pearson Pren-ce Hall. All rights reserved

3 6

6-‐2 © 2010 Pearson Pren-ce Hall. All rights

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Sec-on 6.1 Probability Rules


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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.


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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.


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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.


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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.

(b) The number of leaves on a randomly selected Oak tree.

(c) The length of -me between calls to 911.

EXAMPLE Distinguishing Between Discrete and Continuous Random Variables

Discrete; x = 0, 1, 2, …, 10

Discrete; x = 0, 1, 2, …

Con-nuous; t > 0


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Determine whether the random variable is discrete or continuous.

The number of songs on an MP3 player

A.  Discrete

B.  Continuous

Slide 6-‐ 8 Copyright © 2010 Pearson Educa-on, Inc.

Determine whether the random variable is discrete or continuous.

The number of songs on an MP3 player

A.  Discrete

B.  Continuous



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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.


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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)

0 0.06

1 0.58

2 0.22

3 0.10

4 0.03

5 0.01

EXAMPLE A Discrete Probability Distribu-on


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EXAMPLE Iden)fying Probability Distribu)ons

x P(x)

0 0.16

1 0.18

2 0.22

3 0.10

4 0.30

5 0.01

Is the following a probability distribu-on?


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x P(x)

0 0.16

1 0.18

2 0.22

3 0.10

4 0.30

5 -‐0.01



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x P(x)

0 0.16

1 0.18

2 0.22

3 0.10

4 0.30

5 0.04



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Determine the required value of the missing probability to make the distribution a discrete probability distribution.

A.  0.25

B.  0.65

C.  0.15

D.  0.35

x P(x) 0 0.25 1 0.30 2 ? 3 0.10



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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.


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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.

EXAMPLE Drawing a Probability Histogram

x P(x)

0 0.06

1 0.58

2 0.22

3 0.10

4 0.03

5 0.01


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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.

EXAMPLE Compu-ng the Mean of a Discrete Random Variable

x P(x)

0 0.06

1 0.58

2 0.22

3 0.10

4 0.03

5 0.01


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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.


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As the number of trials of the experiment increases, the mean number of rentals approaches the mean of the probability distribu-on.


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x represents the number of computers in a household.

Find the mean.

A.  1.5

B.  1.4

C.  6

D.  0.3

x P(x) 0 0.15 1 0.45 2 0.30 3 0.10


x represents the number of computers in a household.

Find the mean.

A.  1.5

B.  1.4

C.  6

D.  0.3


x P(x) 0 0.15 1 0.45 2 0.30 3 0.10


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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.


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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 0.99791. Compute the expected value of this policy to the insurance company.

x P(x)

530 0.99791

530 – 250,000 = -‐249,470

0.00209

Survives

Does not survive

E(X) = 530(0.99791) + (-249,470)(0.00209)

= $7.50


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Sec-on 6.2 The Binomial Probability Distribu-on


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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.

(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.

(c) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded.

EXAMPLE Iden)fying Binomial Experiments

Binomial experiment

Not a binomial experiment – not a fixed number of trials.

Not a binomial experiment – the trials are not independent.


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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



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According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time percentage of 79.0% in May, 2008. 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.

EXAMPLE Construc-ng a Binomial Probability Distribu-on


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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?


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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.  0.1598

B.  0.0147

C.  0.3333

D.  0.3144



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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.

EXAMPLE Finding the Mean and Standard Devia)on of a Binomial Random Variable


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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.  2.15

B.  8.55

C.  6.45

D.  2.85


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.  0.0194

B.  0.1126

C.  0.0003

D.  0.0390


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.


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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 ?

EXAMPLE Using the Mean, Standard Devia)on and Empirical Rule to Check for Unusual Results in a Binomial Experiment

The result is unusual since 162 > 159.1


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chapter 36rlhawkmath.pbworks.com/w/file/fetch/65341448/ch6 rl.pdf · 2020. 11. 29. · chapter...

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