Statistics Unlocking the Power of Data

Robin H. Lock, Patti Frazer Lock, Kari Lock Morgan

Chapter 11

Probability Basics - all with Video Answers

Educators

Section 1

Probability Rules

Problem 1

In Exercises $\mathrm{P} .1$ to $\mathrm{P} .7,$ use the information that, for events $\mathrm{A}$ and $\mathrm{B},$ we have $P(A)=0.4, P(B)=0.3,$ and $P(A$ and $B)=0.1.$
Find $P(\operatorname{not} A)$,

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Use the information that, for events $\mathrm{A}$ and $\mathrm{B},$ we have $P(A)=0.4, P(B)=0.3,$ and $P(A$ and $B)=0.1.$
Find $P(\operatorname{not} B).$

Lucas Finney

Numerade Educator

Problem 3

Use the information that, for events $\mathrm{A}$ and $\mathrm{B},$ we have $P(A)=0.4, P(B)=0.3,$ and $P(A$ and $B)=0.1.$
Find $P(A$ or $B).$

Lucas Finney

Numerade Educator

Problem 4

Use the information that, for events $\mathrm{A}$ and $\mathrm{B},$ we have $P(A)=0.4, P(B)=0.3,$ and $P(A$ and $B)=0.1.$
Find $P(A$ if $B).$

Lucas Finney

Numerade Educator

Problem 5

Use the information that, for events $\mathrm{A}$ and $\mathrm{B},$ we have $P(A)=0.4, P(B)=0.3,$ and $P(A$ and $B)=0.1.$
Find $P(B$ if $A).$

Lucas Finney

Numerade Educator

Problem 6

Use the information that, for events $\mathrm{A}$ and $\mathrm{B},$ we have $P(A)=0.4, P(B)=0.3,$ and $P(A$ and $B)=0.1.$
Are events $\mathrm{A}$ and $\mathrm{B}$ disjoint?

Lucas Finney

Numerade Educator

Problem 7

Use the information that, for events $\mathrm{A}$ and $\mathrm{B},$ we have $P(A)=0.4, P(B)=0.3,$ and $P(A$ and $B)=0.1.$
Are events $A$ and $B$ independent?

Lucas Finney

Numerade Educator

Problem 8

In Exercises $\mathrm{P} .8$ to $\mathrm{P} .14$, use the information that, for events $\mathrm{A}$ and $\mathrm{B}$, we have $P(A)=0.8, P(B)=0.4$, and $P(A$ and $B)=0.25.$
Find $P(\operatorname{not} A).$

Lucas Finney

Numerade Educator

Problem 9

Use the information that, for events $\mathrm{A}$ and $\mathrm{B}$, we have $P(A)=0.8, P(B)=0.4$, and $P(A$ and $B)=0.25.$
Find $P(\operatorname{not} B).$

Lucas Finney

Numerade Educator

Problem 10

Use the information that, for events $\mathrm{A}$ and $\mathrm{B}$, we have $P(A)=0.8, P(B)=0.4$, and $P(A$ and $B)=0.25.$
Find $P(A$ or $B)$.

Lucas Finney

Numerade Educator

Problem 11

Use the information that, for events $\mathrm{A}$ and $\mathrm{B}$, we have $P(A)=0.8, P(B)=0.4$, and $P(A$ and $B)=0.25.$
Find $P(A$ if $B)$.

Lucas Finney

Numerade Educator

Problem 12

Use the information that, for events $\mathrm{A}$ and $\mathrm{B}$, we have $P(A)=0.8, P(B)=0.4$, and $P(A$ and $B)=0.25.$
Find $P(B$ if $A)$.

Lucas Finney

Numerade Educator

Problem 13

Use the information that, for events $\mathrm{A}$ and $\mathrm{B}$, we have $P(A)=0.8, P(B)=0.4$, and $P(A$ and $B)=0.25.$
Are events $\mathrm{A}$ and $\mathrm{B}$ disjoint?

Lucas Finney

Numerade Educator

Problem 14

Use the information that, for events $\mathrm{A}$ and $\mathrm{B}$, we have $P(A)=0.8, P(B)=0.4$, and $P(A$ and $B)=0.25.$
Are events A and B independent?

Lucas Finney

Numerade Educator

Problem 15

In Exercises $\mathrm{P} .15$ to $\mathrm{P} .18$, use the fact that we have independent events $\mathrm{A}$ and $\mathrm{B}$ with $P(A)=0.7$ and $P(B)=0.6$.
Find $P(A$ if $B)$.

Lucas Finney

Numerade Educator

Problem 16

Use the fact that we have independent events $\mathrm{A}$ and $\mathrm{B}$ with $P(A)=0.7$ and $P(B)=0.6$.
Find $P(B$ if $A)$.

Lucas Finney

Numerade Educator

Problem 17

Use the fact that we have independent events $\mathrm{A}$ and $\mathrm{B}$ with $P(A)=0.7$ and $P(B)=0.6$.
Find $P(A$ and $B)$.

Lucas Finney

Numerade Educator

Problem 18

Use the fact that we have independent events $\mathrm{A}$ and $\mathrm{B}$ with $P(A)=0.7$ and $P(B)=0.6$.
Find $P(A$ or $B)$.

Lucas Finney

Numerade Educator

Problem 19

Table $\mathrm{P} .3$ gives probabilities for various combinations of events $\mathrm{A}, \mathrm{B},$ and their complements. Use the information from this table in Exercises $\mathrm{P} .19$ to $\mathrm{P} .26 .$.
Find $P(A)$.

Lucas Finney

Numerade Educator

Problem 20

Gives probabilities for various combinations of events $\mathrm{A}, \mathrm{B},$ and their complements. Use the information from this table.
Find $P(\operatorname{not} B)$.

Lucas Finney

Numerade Educator

Problem 21

Gives probabilities for various combinations of events $\mathrm{A}, \mathrm{B},$ and their complements. Use the information from this table.
Find $P(A$ and $B)$.

Lucas Finney

Numerade Educator

Problem 22

Gives probabilities for various combinations of events $\mathrm{A}, \mathrm{B},$ and their complements. Use the information from this table.
Find $P(A$ or $B)$.

Lucas Finney

Numerade Educator

Problem 23

Gives probabilities for various combinations of events $\mathrm{A}, \mathrm{B},$ and their complements. Use the information from this table.
Find $P(A$ if $B)$.

Lucas Finney

Numerade Educator

Problem 24

Gives probabilities for various combinations of events $\mathrm{A}, \mathrm{B},$ and their complements. Use the information from this table.
Find $P(B$ if $A)$.

Lucas Finney

Numerade Educator

Problem 25

Gives probabilities for various combinations of events $\mathrm{A}, \mathrm{B},$ and their complements. Use the information from this table.
Are events $\mathrm{A}$ and $\mathrm{B}$ disjoint?

Lucas Finney

Numerade Educator

Problem 26

Gives probabilities for various combinations of events $\mathrm{A}, \mathrm{B},$ and their complements. Use the information from this table.
Are events $A$ and $B$ independent?

Lucas Finney

Numerade Educator

Problem 27

For Exercises $\mathrm{P} .27$ to $\mathrm{P} .30,$ state whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.)
Draw three skittles (possible colors: yellow, green, red, purple, and orange) from a bag. Let A be the event that all three skittles are green and $\mathrm{B}$ be the event that at least one skittle is red.

Lucas Finney

Numerade Educator

Problem 28

State whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.)
South Africa plays Australia for the championship in the Rugby World Cup. Let $\mathrm{A}$ be the event that Australia wins and $\mathrm{B}$ be the event that South Africa wins. (The game cannot end in a tie.)

Lucas Finney

Numerade Educator

Problem 29

State whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.)
South Africa plays Australia for the championship in the Rugby World Cup. At the same time, Poland plays Russia for the World Team Chess Championship. Let $\mathrm{A}$ be the event that Australia wins their rugby match and $\mathrm{B}$ be the event that Poland wins their chess match.

Lucas Finney

Numerade Educator

Problem 30

State whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.)
Roll two (six-sided) dice. Let $A$ be the event that the first die is a 3 and $B$ be the event that the sum of the two dice is 8

Lucas Finney

Numerade Educator

Problem 31

Each of the following statements demonstrate a common misuse of probability. Explain what is wrong with each statement:
(a) Approximately $10 \%$ of adults are left-handed. So, if we take a simple random sample of 10 adults, 1 of them will be left-handed.
(b) A pitch in baseball can be called a ball or a strike or can be hit by the batter. As there are three possible outcomes, the probability of each is $1 / 3$.
(c) The probability that a die lands with a 1 face up is $1 / 6 .$ So, since rolls of the die are independent, the probability that two consecutive rolls land with a 1 face up is $1 / 6+1 / 6=1 / 3$.
(d) The probability of surviving a heart attack is $2.35 .$

Lucas Finney

Numerade Educator

Problem 32

About $26 \%$ of movies coming out of Hollywood are comedies, Warner Bros has been the lead studio for about $13 \%$ of recent movies, and about $3 \%$ of recent movies are comedies from Warner Bros. $^{2}$ Let $\mathrm{C}$ denote the event a movie is a comedy and $W$ denote the event a movie is produced by Warner Bros.
(a) Write probability expressions for each of the three facts given in the first sentence of the exercise.
(b) What is the probability that a movie is either a comedy or produced by Warner Bros?
(c) What is the probability that a Warner Bros movie is a comedy?
(d) What is the probability that a comedy has Warner Bros as its producer?
(e) What is the probability that a movie coming out of Hollywood is not a comedy?
(f) In terms of movies, what would it mean to say that $\mathrm{C}$ and $\mathrm{W}$ are disjoint events? Are they disjoint events?
(g) In terms of movies, what would it mean to say that $\mathrm{C}$ and $\mathrm{W}$ are independent events? Are they independent events?

Lucas Finney

Numerade Educator

Problem 33

From its founding through $2015,$ the Rock and Roll Hall of Fame has inducted 303 groups or individuals. ${ }^{3}$ Table $\mathrm{P} .4$ shows how many of the inductees have been female or have included female members and also shows
how many of the inductees have been performers. (The full dataset is available in RockandRoll.) Letting Frepresent the event of having female members (or being a female) and MP represent the event of being a (music) performer, write each of the following questions as a probability expression and find the probability.

What is the probability that an inductee chosen at random:
(a) Is a performer?
(b) Does not have any female members?
(c) Has female members if it is a performer?
(d) Is not a performer if it has no female members?
(e) Is a performer with no female members?
(f) Is either not a performer or has female members?

Lucas Finney

Numerade Educator

Problem 34

From its founding through $2016,$ the Hockey Hall of Fame has inducted 268 players. ${ }^{4}$ Table $\mathrm{P} .5$ shows the number of players by place of birth and by position played. If a player is chosen at random from all player inductees into the Hockey Hall of Fame, let $C$ represent the event of being born in Canada and $D$ represent the event of being a defenseman. Write each of the following questions as a probability expression and find the probability.
(a) What is the probability that an inductee chosen at random is Canadian?
(b) What is the probability that an inductee chosen at random is not a defenseman?
(c) What is the probability that a player chosen at random is a defenseman born in Canada?
(d) What is the probability that a player chosen at random is either born in Canada or a defenseman?
(e) What is the probability that a Canadian inductee plays defense?
(f) What is the probability that an inductee who plays defense is Canadian?

Lucas Finney

Numerade Educator

Problem 35

In a bag of peanut $M$ \& M's, there are $80 \mathrm{M} \& \mathrm{Ms}$, with 11 red ones, 12 orange ones, 20 blue ones, 11 green ones, 18 yellow ones, and 8 brown ones. They are mixed up so that each candy piece is equally likely to be selected if we pick one.
(a) If we select one at random, what is the probability that it is red?
(b) If we select one at random, what is the probability that it is not blue?
(c) If we select one at random, what is the probability that it is red or orange?
(d) If we select one at random, then put it back, mix them up well (so the selections are independent) and select another one, what is the probability that both the first and second ones are blue?
(e) If we select one, keep it, and then select a second one, what is the probability that the first one is red and the second one is green?

Danielle Fairburn

Danielle Fairburn

Numerade Educator

Problem 36

As in Exercise $\mathrm{P} .35,$ we have a bag of peanut $\mathrm{M} \& \mathrm{M}$ 's with $80 \mathrm{M} \& \mathrm{Ms}$ in it, and there are 11 red ones, 12 orange ones, 20 blue ones, 11 green ones, 18 yellow ones, and 8 brown ones. They are mixed up so that each is equally likely to be selected if we pick one.
(a) If we select one at random, what is the probability that it is yellow?
(b) If we select one at random, what is the probability that it is not brown?
(c) If we select one at random, what is the probability that it is blue or green?
(d) If we select one at random, then put it back, mix them up well (so the selections are independent) and select another one, what is the probability that both the first and second ones are red?
(e) If we select one, keep it, and then select a second one, what is the probability that the first one is yellow and the second one is blue?

Lucas Finney

Numerade Educator

Problem 37

During the $2015-16$ NBA season, Stephen Curry of the Golden State Warriors had a free throw shooting percentage of 0.908 . Assume that the probability Stephen Curry makes any given free throw is fixed at 0.908 , and that free throws are independent.
(a) If Stephen Curry shoots two free throws, what is the probability that he makes both of them?
(b) If Stephen Curry shoots two free throws, what is the probability that he misses both of them?
(c) If Stephen Curry shoots two free throws, what is the probability that he makes exactly one of them?

Lucas Finney

Numerade Educator

Problem 38

The most common form of color blindness is an inability to distinguish red from green. However, this particular form of color blindness is much more common in men than in women (this is because the genes corresponding to the red and green receptors are located on the X-chromosome). Approximately $7 \%$ of American men and $0.4 \%$ of American women are red-green color-blind. $^{5}$
(a) If an American male is selected at random, what is the probability that he is red-green color-blind?
(b) If an American female is selected at random, what is the probability that she is NOT redgreen color-blind?
(c) If one man and one woman are selected at random, what is the probability that neither are redgreen color-blind?
(d) If one man and one woman are selected at random, what is the probability that at least one of them is red-green color-blind?

Lucas Finney

Numerade Educator

Problem 39

Approximately $7 \%$ of men and $0.4 \%$ of women are red-green color-blind (as in Exercise P.38). Assume that a statistics class has 15 men and 25 women.
(a) What is the probability that nobody in the class is red-green color-blind?
(b) What is the probability that at least one person in the class is red-green color-blind?
(c) If a student from the class is selected at random, what is the probability that he or she will be redgreen color-blind?

Sheryl Ezze

Numerade Educator

Problem 40

The US Social Security Administration collects information on the life expectancy and death rates of the population. Table $\mathrm{P} .6$ gives the number of US men out of 100,000 born alive who will survive to a given age, based on 2011 mortality rates. For example, 50,344 of 100,000 US males live to their 80 th birthday.
(a) What is the probability that a man lives to age $60 ?$
(b) What is the probability that a man dies before age $70 ?$
(c) What is the probability that a man dies at age 90 (after his 90th and before his 91 st birthday)?
(d) If a man lives until his 90 th birthday, what is the probability that he will die at the age of $90 ?$
(e) If a man lives until his 80 th birthday, what is the probability that he will die at the age of $90 ?$
(f) What is the probability that a man dies between the ages of 60 and $89 ?$
(g) If a man lives until his 60 th birthday, what is the probability that he lives to be at least 90 years old?

Check back soon!

Problem 41

The Standard and Poor 500 (S\&P 500 ) is a weighted average of the stocks for 500 large companies in the United States. It is commonly used as a measure of the overall performance of the US stock market. Between January 1,2009 and January $1,2012,$ the S\&P 500 increased for 423 of the 756 days that the stock market was open. We will investigate whether changes to the S\&P 500 are independent from day to day. This is important, because if changes are not independent, we should be able to use the performance on the current day to help predict performance on the next day.
(a) What is the probability that the S\&P 500 increased on a randomly selected market day between January 1,2009 and January $1,2012 ?$
(b) If we assume that daily changes to the $S \& P$ 500 are independent, what is the probability that the S\&P 500 increases for two consecutive days? What is the probability that the S\&P 500 increases on a day, given that it increased the day before?
(c) Between January 1, 2009 and January 1,2012 the S\&P 500 increased on two consecutive market days 234 times out of a possible $755 .$ Based on this information, what is the probability that the S\&P 500 increases for two consecutive days? What is the probability that the S\&P 500 increases on a day, given that it increased the day before?
d) Compare your answers to part (b) and part (c). Do you think that this analysis proves that daily changes to the S\&P 500 are not independent?

Check back soon!

Problem 42

A friend makes three pancakes for breakfast. One of the pancakes is burned on both sides, one is burned on only one side, and the other is not burned on either side. You are served one of the pancakes at random, and the side facing you is burned. What is the probability that the other side is burned? (Hint: Use conditional probability.)

Check back soon!