Introduction to Probability and Statistics

William Mendenhall, III, Robert J. Beaver, Barbara M. Beaver

Chapter 5 Discrete Probability Distributions - all with Video Answers

Educators

Section 1

Discrete Random Variables and Their Probability Distributions

01:31

Problem 1

What are the two requirements for a discrete probability distribution?

Christopher Stanley

Numerade Educator

01:01

Problem 2

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Total number of points scored in a football game

Sanchit Jain

Numerade Educator

01:01

Problem 3

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Shelf life of a particular drug

Sanchit Jain

Numerade Educator

01:01

Problem 4

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Height of the ocean's tide at a given location

Sanchit Jain

Numerade Educator

01:01

Problem 5

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Length of a 2 -year-old black bass

Sanchit Jain

Numerade Educator

01:01

Problem 6

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Number of aircraft near-collisions in a year

Sanchit Jain

Numerade Educator

01:01

Problem 7

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Increase in length of life attained by a cancer patient as a result of surgery

Sanchit Jain

Numerade Educator

01:01

Problem 8

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Tensile breaking strength (in pounds per square inch) of 1 -inch-diameter steel cable

Sanchit Jain

Numerade Educator

01:01

Problem 9

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Number of deer killed per year in a state wildlife preserve

Sanchit Jain

Numerade Educator

01:01

Problem 10

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Number of overdue accounts in a department store at a particular time

Sanchit Jain

Numerade Educator

01:01

Problem 11

Identify the random variables in Exercises $2-11$ as either discrete or continuous.
Your blood pressure

Sanchit Jain

Numerade Educator

01:27

Use the probability distribution for the random variable $x$ to answer the questions in Exercises 12-16.
$$\begin{array}{l|rrrrrr}x & 0 & 1 & 2 & 3 & 4 & 5 \\\hline p(x) & .1 & .3 & .4 & .1 & ? & .05\end{array}$$
Construct a probability histogram to describe $p(x)$.

Wendi Zhao

Numerade Educator

02:11

Problem 14

Use the probability distribution for the random variable $x$ to answer the questions in Exercises 12-16.
$$\begin{array}{l|rrrrrr}x & 0 & 1 & 2 & 3 & 4 & 5 \\\hline p(x) & .1 & .3 & .4 & .1 & ? & .05\end{array}$$
$$
\text { Find } \mu, \sigma^{2}, \text { and } \sigma \text { . }
$$

Wendi Zhao

Numerade Educator

01:48

Problem 15

Use the probability distribution for the random variable $x$ to answer the questions in Exercises 12-16.
$$\begin{array}{l|rrrrrr}x & 0 & 1 & 2 & 3 & 4 & 5 \\\hline p(x) & .1 & .3 & .4 & .1 & ? & .05\end{array}$$
Locate the interval $\mu \pm 2 \sigma$ on the $x$ -axis of the histogram. What is the probability that $x$ will fall into this interval?

Wendi Zhao

Numerade Educator

01:34

Problem 16

Use the probability distribution for the random variable $x$ to answer the questions in Exercises 12-16.
$$\begin{array}{l|rrrrrr}x & 0 & 1 & 2 & 3 & 4 & 5 \\\hline p(x) & .1 & .3 & .4 & .1 & ? & .05\end{array}$$
If you were to select a very large number of values of $x$ from the population, would most fall into the interval $\mu \pm 2 \sigma ?$ Explain.

Wendi Zhao

Numerade Educator

01:14

Problem 17

Use the probability distribution for the random variable $x$ to answer the questions in Exercises 17-21.
$$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\hline p(x) & .1 & .3 & .3 & ? & .1\end{array}$$
Find $p(3)$

Wendi Zhao

Numerade Educator

01:28

Problem 18

Use the probability distribution for the random variable $x$ to answer the questions.
$$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\hline p(x) & .1 & .3 & .3 & ? & .1\end{array}$$
Construct a probability histogram for $p(x)$.

Wendi Zhao

Numerade Educator

01:41

Problem 19

Use the probability distribution for the random variable $x$ to answer the questions.
$$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\hline p(x) & .1 & .3 & .3 & ? & .1\end{array}$$
Calculate the population mean, variance, and standard deviation.

Wendi Zhao

Numerade Educator

01:08

Problem 20

Use the probability distribution for the random variable $x$ to answer the questions.
$$\begin{array}{l|lllll}x & 0 & 1 & 2 & 3 & 4 \\\hline p(x) & .1 & .3 & .3 & ? & .1\end{array}$$
What is the probability that $x$ is greater than $2 ?$

Wendi Zhao

Numerade Educator

01:25

Problem 21

Wendi Zhao

Numerade Educator

01:50

Problem 22

For the random variables described in Exercises $22-26,$ find and graph the probability distribution for $x .$ Then calculate the mean, variance, and standard deviation.
Let $x$ be the number observed on the throw of a single balanced die.

Wendi Zhao

Numerade Educator

03:18

Problem 23

For the random variables described, find and graph the probability distribution for $x .$ Then calculate the mean, variance, and standard deviation.
Toss a pair of dice and record $x,$ the sum of the numbers on the two upper faces.

Wendi Zhao

Numerade Educator

03:39

Problem 24

For the random variables described, find and graph the probability distribution for $x .$ Then calculate the mean, variance, and standard deviation.
Of adults 18 years and older, $47 \%$ admit to texting while driving. ' Three adults are randomly selected and $x$, the number who admit to texting while driving is recorded.

Wendi Zhao

Numerade Educator

02:15

Problem 25

For the random variables described, find and graph the probability distribution for $x .$ Then calculate the mean, variance, and standard deviation.
Five applicants have applied for two positions: two women and three men. All are equally qualified and there is no preference for choosing either gender. Let $x$ be the number of women chosen to fill the two positions.

Wendi Zhao

Numerade Educator

02:31

Problem 26

For the random variables described, find and graph the probability distribution for $x .$ Then calculate the mean, variance, and standard deviation.
A piece of electronic equipment contains 6 computer chips, two of which are defective. Three chips are randomly selected and inspected, and $x$, the number of defective chips in the selection is recorded.

Wendi Zhao

Numerade Educator

01:15

Problem 27

Let $x$ represent the number of times a customer visits a grocery store in a 1 -week period. Assume this is the probability distribution of $x$ :
$$\begin{array}{l|cccc}x & 0 & 1 & 2 & 3 \\\hline p(x) & .1 & .4 & .4 & .1\end{array}$$
Find the expected value of $x$, the average number of times a customer visits the store.

Wendi Zhao

Numerade Educator

06:52

Problem 28

A key ring contains four office keys that are identical in appearance, but only one will open your office door. Suppose you randomly select one key and try it. If it does not fit, you randomly select one of the three remaining keys. If that key does not fit, you randomly select one of the last two. Each different sequence that could occur in selecting the keys represents a set of equally likely simple events.
a. List the simple events in $S$ and assign probabilities to the simple events.
b. Let $x$ equal the number of keys that you try before you find the one that opens the door $(x=1,2,3,4)$. Then assign the appropriate value of $x$ to each simple event.
c. Calculate the values of $p(x)$ and display them in a table.
d. Construct a probability histogram for $p(x)$.

Ahmad Reda

Numerade Educator

02:48

Problem 29

Wells Past experience has shown that, on the average, only 1 in 10 wells drilled hits oil. Let $x$ be the number of drillings until the first success (oil is struck). Assume that the drillings represent independent events.
a. Find $p(1), p(2),$ and $p(3)$.
b. Give a formula for $p(x)$.
c. Graph $p(x)$

Wendi Zhao

Numerade Educator

03:16

Problem 30

In a county containing a large number of rural homes, $60 \%$ of the homes are insured against fire. Four rural homeowners are chosen at random from this county, and $x$ are found to be insured against fire. Find the probability distribution for $x$.
What is the probability that at least three of the four will be insured?

Wendi Zhao

Numerade Educator

00:36

Problem 31

A roulette wheel contains 38 pocketsthe numbers 1 through $36,0,$ and $00 .$ The wheel is spun and the "winning" pocket is recorded, with any one pocket just as likely as any other. Suppose you bet $\$ 5$ on the number 18 . The payoff on this type of bet is usually $\$ 35$ for a $\$ 1$ bet. What is your expected gain?

James Macpherson

Numerade Educator

01:01

Problem 32

A fire-detection device uses three temperature-sensitive cells acting independently of one another so that any one or more can activate the alarm. Each cell has a probability $p=.8$ of activating the alarm when the temperature reaches $57^{\circ} \mathrm{C}$ or higher. Let $x$ equal the number of cells activating the alarm when the temperature reaches $57^{\circ} \mathrm{C}$.
a. Find the probability distribution of $x$.
b. Find the probability that the alarm will function when the temperature reaches $57^{\circ} \mathrm{C}$.
c. Find the expected value and the variance for the random variable $x$.

Wendi Zhao

Numerade Educator

01:52

Problem 33

You can insure a $\$ 50,000$ diamond for its total value by paying a premium of $D$ dollars. If the probability of loss in a given year is estimated to be .01, what premium should the insurance company charge if it wants the expected gain to equal $\$ 1000 ?$

Lynn Larson

Numerade Educator

02:36

Problem 34

The maximum patent life for a new drug is 17 years. Subtracting the length of time required by the FDA for testing and approval of the drug provides the actual patent life of the drug- that is, the length of time that a company has to recover research and development costs and make a profit. Suppose the distribution of the lengths of patent life for new drugs is as shown here:
$$\begin{array}{l|lllllllllll}\text { Years, } x & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\\hline p(x) & .03 & .05 & .07 & .10 & .14 & .20 & .18 & .12 & .07 & .03 & .01\end{array}$$
a. Find the expected number of years of patent life for a new drug.
b. Find the standard deviation of $x$.
c. Find the probability that $x$ falls into the interval $\mu \pm 2 \sigma$

Wendi Zhao

Numerade Educator

04:13

Problem 35

Most coffee drinkers take a little time each day for their favorite beverage, and many take more than one coffee break every day. The following table, adapted from a USA Today snapshot, shows the probability distribution for $x,$ the number of coffee breaks taken per day by coffee drinkers.
$$\begin{array}{l|llllll}x & 0 & 1 & 2 & 3 & 4 & 5 \\\hline p(x) & .28 & .37 & .17 & .12 & .05 & .01\end{array}$$
a. What is the probability that a randomly selected coffee drinker would take no coffee breaks during the day?
b. What is the probability that a randomly selected coffee drinker would take more than two coffee breaks during the day?
c. Calculate the mean and standard deviation for the random variable $x$.
d. Find the probability that $x$ falls into the interval $\mu \pm 2 \sigma$

Bon Zapata

Numerade Educator

01:20

Problem 36

A shipping company knows that the cost of delivering a small package within 24 hours is $\$ 14.80 .$ The company charges $\$ 15.50$ for shipment but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only $2 \%$ of its packages within the 24 -hour period, what is the expected gain per package?

Prabhakar Kumar

Numerade Educator

01:23

Problem 37

A CEO is considering buying an insurance policy to cover possible losses incurred by marketing a new product. If the product is a complete failure, a loss of $\$ 800,000$ would be incurred; if it is only moderately successful, a loss of $\$ 250,000$ would be incurred. Insurance actuaries have determined that the probabilities that the product will be a failure or only moderately successful are .01 and $.05,$ respectively. Assuming that the $\mathrm{CEO}$ is willing to ignore all other possible losses, what premium should the insurance company charge for a policy in order to break even?

Hast Aggarwal

Numerade Educator

03:21

Problem 38

The board of directors of a major symphony orchestra has voted to create a committee to handle employee complaints. The committee will consist of the president and vice president of the symphony board and two orchestra representatives. The two orchestra representatives will be randomly selected from a list of six volunteers, consisting of four men and
two women.
a. Find the probability distribution for $x,$ the number of women chosen to be orchestra representatives.
b. What is the probability that both orchestra representatives will be women?
c. Find the mean and variance for the random variable $x$.

Joshua Eastwood

Numerade Educator

Introduction to Probability and Statistics

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