Chapter 4, Elementary Probability Theory Video Solutions, Understandable Statistics, Concepts and Methods

01:10

Problem 1

List three methods of assigning probabilities.

Harsh Gadhiya

Numerade Educator

01:13

Problem 2

Suppose the newspaper states that the probability of rain today is $30 \% .$ What is the complement of the event "rain today"? What is the probability of the complement?

Harsh Gadhiya

Numerade Educator

01:16

Problem 3

What is the probability of
(a) an event $A$ that is certain to occur?
(b) an event $B$ that is impossible?

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Numerade Educator

01:20

Problem 4

What is the law of large numbers? If you were using the relative frequency of an event to estimate the probability of the event, would it be better to use 100 trials or 500 trials? Explain.

Harsh Gadhiya

Numerade Educator

00:50

A Harris Poll indicated that of those adults who drive and have a cell phone, the probability that a driver between the ages of 18 and 24 sends or reads text messages is $0.51 .$ Can this probability be applied to all drivers with cell phones? Explain.

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Numerade Educator

03:09

Problem 6

According to a recent Harris Poll of adults with pets, the probability that the pet owner cooks especially for the pet either frequently or occasionally is 0.24.
(a) From this information, can we conclude that the probability a male owner cooks for the pet is the same as for a female owner? Explain.
(b) According to the poll, the probability a male owner cooks for his pet is 0.27 whereas the probability a female owner does so is 0.22. Let's explore how such probabilities might occur. Suppose the pool of pet owners surveyed consisted of 200 pet owners, 100 of whom are male and 100 of whom are female. Of the pet owners, a total of 49 cook for their pets. Of the 49 who cook for their pets, 27 are male and 22 are female. Use relative frequencies to determine the probability a pet owner cooks for a pet, the probability a male owner cooks for his pet, and the probability a female owner cooks for her pet.

Harsh Gadhiya

Numerade Educator

01:23

Problem 7

A recent Harris Poll survey of 1010 U.S. adults selected at random showed that 627 consider the occupation of firefighter to have very great prestige. Estimate the probability (to the nearest hundredth) that a U.S. adult selected at random thinks the occupation of firefighter has very great prestige.

Harsh Gadhiya

Numerade Educator

01:05

Problem 8

What is the probability that a day of the week selected at random will be a Wednesday?

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Numerade Educator

01:32

Problem 9

An investment opportunity boasts that the chance of doubling your money in 3 years is $95 \% .$ However, when you research the details of the investment, you estimate that there is a $3 \%$ chance that you could lose the entire investment. Based on this information, are you certain to make money on this investment? Are there risks in this investment opportunity?

Harsh Gadhiya

Numerade Educator

01:50

Problem 10

A sample space consists of 4 simple events: $A, B, C, D$ Which events comprise the complement of $A ?$ Can the sample space be viewed as having two events, $A$ and $A^{\text {c }}$ ? Explain.

Harsh Gadhiya

Numerade Educator

03:03

Problem 11

Consider a family with 3 children. Assume the probability that one child is a boy is 0.5 and the probability that one child is a girl is also
$0.5,$ and that the events "boy" and "girl" are independent.
(a) List the equally likely events for the gender of the 3 children, from oldest
to youngest.
(b) What is the probability that all 3 children are male? Notice that the complement of the event "all three children are male" is "at least one of the children is female." Use this information to compute the probability that at least one child is female.

Harsh Gadhiya

Numerade Educator

02:34

Problem 12

Consider the experiment of tossing a fair coin 3 times. For each coin, the possible outcomes are heads or tails.
(a) List the equally likely events of the sample space for the three tosses.
(b) What is the probability that all three coins come up heads? Notice that the complement of the event "3 heads" is "at least one tail." Use this information to compute the probability that there will be at least one tail.

Harsh Gadhiya

Numerade Educator

01:12

Problem 13

On a single toss of a fair coin, the probability of heads is 0.5 and the probability of tails is $0.5 .$ If you toss a coin twice and get heads on the first toss, are you guaranteed to get tails on the second toss? Explain.

Harsh Gadhiya

Numerade Educator

03:11

Problem 14

(a) Explain why -0.41 cannot be the probability of some event.
(b) Explain why 1.21 cannot be the probability of some event.
(c) Explain why $120 \%$ cannot be the probability of some event.
(d) Can the number 0.56 be the probability of an event? Explain.

Harsh Gadhiya

Numerade Educator

01:52

Problem 15

Probability Estimate: Wiggle Your Ears Can you wiggle your ears? Use the students in your statistics class (or a group of friends) to estimate the percentage of people who can wiggle their ears. How can your result be thought of as an estimate for the probability that a person chosen at random can wiggle his or her ears? Comment: National statistics indicate that about $13 \%$ of Americans can wiggle their ears (Source: Bernice Kanner, Are You Normal?, St. Martin's Press, New York).

Harsh Gadhiya

Numerade Educator

01:24

Problem 16

Can you raise one eyebrow at a time? Use the students in your statistics class (or a group of friends) to estimate the percentage of people who can raise one eyebrow at a time. How can your result be thought of as an estimate for the probability that a person chosen at random can raise one eyebrow at a time? Comment: National statistics indicate that about $30 \%$ of Americans can raise one eyebrow at a time (see source in Problem 15 ).

Harsh Gadhiya

Numerade Educator

04:07

Problem 17

Myers-Briggs: Personality Types Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates? The following data give an indication .
Suppose that a married couple is selected at random.
(a) Use the data to estimate the probability that they will have $0,1,2,3,$ or 4 personality preferences in common.
(b) Do the probabilities add up to $1 ?$ Why should they? What is the sample space in this problem?

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Numerade Educator

02:24

Problem 18

(a) If you roll a single die and count the number of dots on top, what is the sample space of all possible outcomes? Are the outcomes equally likely?
(b) Assign probabilitics to the outcomes of the sample space of part (a). Do the probabilities add up to $1 ?$ Should they add up to $1 ?$ Explain.
(c) What is the probability of getting a number less than 5 on a single throw?
(d) What is the probability of getting 5 or 6 on a single throw?

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Numerade Educator

01:12

Problem 19

Creativity When do creative people get their best ideas? USA Today did a survey of 966 inventors (who hold U.S. patents) and obtained the following information:
(a) Assuming that the time interval includes the left limit and all the times up to but not including the right limit, estimate the probability that an inventor has a best idea during each time interval: from 6 A.M. to 12 noon, from
12 noon to 6 P.M., from 6 P.M. to 12 midnight, from 12 midnight to 6 A.M.
(b) Do the probabilities of part (a) add up to 1? Why should they? What is the sample space in this problem?

Harsh Gadhiya

Numerade Educator

02:48

Problem 20

A botanist has developed a new hybrid cotton plant that can withstand insects better than other cotton plants. However, there is some concern about the germination of seeds from the new plant. To estimate the probability that a seed from the new plant will germinate, a random sample of 3000 seeds was planted in warm, moist soil. Of these seeds, 2430 germinated.
(a) Use relative frequencies to estimate the probability that a seed will germinate. What is your estimate?
(b) Use relative frequencies to estimate the probability that a seed will not germinate. What is your estimate?
(c) Either a seed germinates or it does not. What is the sample space in this problem? Do the probabilities assigned to the sample space add up to $1 ?$ Should they add up to $1 ?$ Explain.
(d) Are the outcomes in the sample space of part (c) equally likely?

Harsh Gadhiya

Numerade Educator

03:32

Problem 21

Sometimes probability statements are expressed in terms of odds.
The odds in favor of an event $A$ are the ratio $\frac{P(A)}{P(n o t A)}=\frac{P(A)}{P\left(A^{c}\right)}$
For instance, if $P(A)=0.60,$ then $P\left(A^{C}\right)=0.40$ and the odds in favor of $A$ are
$$
\frac{0.60}{0.40}=\frac{6}{4}=\frac{3}{2}, \text { written as } 3 \text { to } 2 \text { or } 3: 2
$$
(a) Show that if we are given the odds in favor of event $A$ as $n: m,$ the probability of event $A$ is given by $P(A)=\frac{n}{n+m} .$ Hint: Solve the equation
$$
\frac{n}{m}=\frac{P(A)}{1-P(A)} \text { for } P(A)
$$
(b) A telemarketing supervisor tells a new worker that the odds of making a sale on a single call are 2 to $15 .$ What is the probability of a successful call?
(c) A sports announcer says that the odds a basketball player will make a free throw shot are 3 to $5 .$ What is the probability the player will make the shot?

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Numerade Educator

08:15

Problem 22

Betting odds are usually stated against the event happening (against winning).
The odds against event $W$ are the ratio $\frac{P(\text {not} W)}{P(W)}=\frac{P\left(W^{*}\right)}{P(W)}$
In horse racing, the betting odds are based on the probability that the horse does not win.
(a) Show that if we are given the odds against an event $W$ as $a: b,$ the probability of not $W$ is $P\left(W^{\circ}\right)=\frac{a}{a+b} .$ Hint: Solve the cquation $\frac{a}{b}=\frac{P(W)}{1-P\left(W^{\circ}\right)}$ for $P\left(W^{c}\right)$
(b) In a recent Kentucky Derby, the betting odds for the favorite horse, Point Given, were 9 to $5 .$ Use these odds to compute the probability that Point Given would lose the race. What is the probability that Point Given would win the race?
(c) In the same race, the betting odds for the horse Monarchos were 6 to 1 . Use these odds to estimate the probability that Monarchos would lose the race. What is the probability that Monarchos would win the race?
(d) Invisible Ink was a long shot, with betting odds of 30 to $1 .$ Use these odds to estimate the probability that Invisible Ink would lose the race. What is the probability the horse would win the race? For further information on the Kentucky Derby, visit the web site of the Kentucky Derby.

Harsh Gadhiya

Numerade Educator

02:59

Problem 23

Customers John runs a computer software store. Yesterday he counted 127 people who walked by his store, 58 of whom came into the store.
Of the $58,$ only 25 bought something in the store.
(a) Estimate the probability that a person who walks by the store will enter the store.
(b) Estimate the probability that a person who walks into the store will buy something.
(c) Estimate the probability that a person who walks by the store will come in and buy something.
(d) Estimate the probability that a person who comes into the store will buy nothing.

Harsh Gadhiya

Numerade Educator

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