Homework 9 – Solutions
Chapter 7A
Review of the Multiplication Principle. Use the skills covered in the Brief Review on
p. 420 to answer the following questions.
13. How many choices of car do you have if a particular model comes in 15 colors and
three styles (sedan, station wagon, or hatchback)?
There are 15 × 3 = 45 choices.
18. You are required to take five courses, one each in humanities, sociology, science, math,
and music. You have a choice of four humanities courses, three sociology courses, five
science courses, two math courses, and three music courses. How many different sets
of five courses are possible?
There are 4 × 3 × 5 × 2 × 3 = 360 sets of five courses.
Theoretical Probabilities. Use the theoretical method to determine the probability of
the given outcomes and events. State any assumptions that you make.
24. Rolling two dice and getting a sum of 4.
There are 6 × 6 = 36 possible outcomes, three of which sum to 4, so assuming all
outcomes are equally likely to occur the probability is 3/36 = 1/12.
35. Randomly selecting a three-child family with three girls.
There are 8 possible outcomes with a 3-child family, one of which is a family with
three girls, so the probability is 1/8.
36. Randomly selecting a pair of white socks from a drawer that holds five pairs of black
socks, nine pairs of blue socks, and eight pairs of white socks.
There are 5+9+8 = 22 pairs of socks, eight of which are white, so assuming each pair
is equally likely to be selected the probability of selecting a white pair is 8/22 = 4/11.
Empirical Probabilities. Use the empirical method to estimate the following probabilities.
42. You count 42 heads when you toss a coin 100 times. If you don’t know whether the
coin is fair, what is the probability that the next toss will be a tail?
Not knowing whether the coin is fair we should use the empirical probability of a
head, 42/100, so the probability of a tail is 1 − (42/100) = 58/100 = .58.
Event Not Occurring. Determine the probability of the following events. State any assumptions you use.
48. What is the probability of not tossing three tails with three fair coins?
There are 8 possible outcomes when tossing three fair coins, only one of which gives
three tails, so the probability of tossing three tails is 1/8 and the probability of not
tossing three heads is 1 − (1/8) = 7/8.
Odds. Use the definition given in the text to find both the odds for and the odds against
the following events.
57. Rolling a fair die and getting a 1 or a 2.
The probability of getting a 1 or a 2 is 2/6 = 1/3, and the probability of not getting
a 1 or a 2 is 4/6 = 2/3. Thus the odds for the event are (1/3) ÷ (2/3) = 1 to 2, and
the odds against the event are (2/3) ÷ (1/3) = 2 to 1.
63. Gender Politics. The following table gives the gender and political party of the 100
delegates at a political convention. Suppose you encounter a delegate at random.
Republicans
Democrats
Independents
Women
21
25
6
Men
28
16
4
a. What is the probability that you meet a woman?
There are 52 women at the convention, so the probability that a random delegate
is a women is 52/100 = 0.52.
b. What is the probability that you meet an Independent?
There are 10 Independents, so the probability that a random delegate is an Independent is 10/100 = 0.1.
c. What is the probability that you do not meet a Democrat?
There are 10 Democrats, and thus 59 non-Democrats, so the probability that a
random delegate is not a Democrat is 59/100 = 0.59.
d. What is the probability that you meet a female Republican?
There are 21 female Republicans, so the probability is 21/100 = 0.21.
e. What is the probability that you meet someone who is not a male Democrat?
There are 16 male Democrats, so the probability of meeting one is 16/100 = 0.16.
Thus the probability of meeting someone who is not a male Democrat is 1−0.16 =
0.84.
Chapter 7B
Does it make sense? Decide whether each of the following statements makes sense (or is
clearly true) or does not make sense (or is clearly false). Explain your reasoning.
5. The probability of getting heads and tails when you toss a coin is 0, but the probability
of getting heads or tails is 1.
Makes sense. You cannot get both heads and tails on a single toss of a coin, but you
are guaranteed to get one or the other.
And Probabilities. Determine whether the following events are independent or dependents. Then find the probability of the event.
17. Rolling four fair dice and getting an even number on all four dice.
The events are independent. Because the probability of rolling an even number on
each throw is 1/2, the probability of getting an even number on all four dice is
(1/2)4 = 1/16.
22. Randomly selecting a three-person committee consisting entirely of Americans from a
pool of 12 British people and 18 Americans.
The events are dependent. The probability is (18/30) × (17/29) × (16/28) ≈ 0.201.
Either/Or Probabilities. Determine whether the following events are overlapping or nonoverlapping. Then find the probability of the event.
28. Getting a sum of either 6 or 8 on a roll of two dice.
The events are non-overlapping, so P (6 or 8) = P (6) + P (8) = (5/36) + (5/36) =
5/18 ≈ 0.278.
31. Getting a sum of either 10, 11, or 12 on a roll of two dice.
The events are non-overlapping, so P (10 or 11 or 12) = P (10) + P (11) + P (12) =
(3/36) + (2/36) + (1/36) = 1/6 ≈ 0.167.
34. Rolling a 3 or a red number on die on which the even numbers are red and the odd
numbers are black.
The events are non-overlapping, so P (3 or red) = P (3) + P (red) = (1/6) + (3/6) =
4/6 = 2/3 ≈ 0.667.
At Least Once Problems. Use the at least once rule to find the probabilities of the following events.
42. Getting at least one head when tossing six fair coins.
P (at least one head) = 1 − P (six tails) = 1 − (1/2)6 ≈ 0.984.
48. Meeting at least one left-handed person in eight random encounters on campus when
the incidence rate is 11% (11 in 100 people are left-handed).
P (at least one left-handed) = 1 − P (eight right-handed) = 1 − (0.89)8 ≈ 0.606.
53. Polling Calls. A telephone pollster has a list of names and telephone numbers for 45
voters, 20 of whom are listed as registered Democrats and 25 of whom are listed as
registered Republicans. Calls are made in random order. Suppose you want to find
the probability that the first two calls are to Republicans.
a. Are these independent or dependent events? Explain.
The events are dependent because there are fewer people to choose from on each
subsequent call.
b. If you treat them as dependent events, what is the probability that the first two
calls are to Republicans?
P (R1 and R2 ) = P (R1 ) × P (R2 given R1 ) = (25/45) × (24/44) ≈ 0.303.
c. If you treat them as independent events, what is the probability that the first two
calls are to Republicans?
P (R1 and R2 ) = P (R1 ) × P (R2 ) = (25/45) × (25/45) ≈ 0.309.
d. Compare the results of parts b and c.
Because the list of names is relatively large and removing one person from the list
doesn’t appreciably change the probability, the two results are fairly similar.
58. Sexually Transmitted Infections. It is estimated that by age 24, one in three sexually active people will have a sexually transmitted infection (STI). Assume that 20%
of the sexually active students on a particular campus have an STI.
a. If a student has ten sexual partners over a period of time, what is the probability
that a least one of these partners is infected with an STI?
P (at least one STI) = 1 − (0.8)10 ≈ 0.89.
b. If a student has 20 different sexual partners over a period of time what is the
probability that a least one of these partners is infected with an STI?
P (at least one STI) = 1 − (0.8)20 ≈ 0.99.