CHAPTER 13 + 11
BINOMIAL DISTRIBUTIONS +
1. A small school club has 16 students with 12 males and 4 females. Two representatives are needed to meet
with the principal. The names of the 16 students are put in a hat, and 2 are selected at random to represent the
club. Let X be the number of males selected. Then X has
a. a binomial distribution, with mean 0.75.
b. a binomial distribution, with mean 1.5.
c. a binomial distribution, with 12 trials.
d. none of the above.
2. A small class has 10 students. Five of the students are male and 5 are female. I write the name of each
student on a 3 by 5 card. The cards are shuffled thoroughly and I choose one at random, observe the name of the
student, and replace it in the set. The cards are thoroughly reshuffled and I again choose a card at random,
observe the name, and replace it in the set. This is done a total of four times. Let X be the number of cards
observed in these four trials with a name corresponding to a male student. The random variable X has which of
the following probability distributions?
a. The normal distribution, with mean 2 and variance 1
b. The binomial distribution, with parameters n = 4 and p = 0.5
c. The binomial distribution, with parameters n = 10 and p = 0.5
d. none of the above
3. For which of the following counts would a binomial probability model be reasonable?
a. the number of phone calls received in a one hour period
b. the number of hearts in a hand of 5 cards dealt from a standard deck of 52 cards that has been thoroughly
shuffled
c. the number of sevens in a randomly selected set of five random digits from your table of random digits
(Table B)
d. all if the above
4. A set of 10 cards consists of 5 red cards and 5 black cards. The cards are shuffled thoroughly and I choose
one at random, observe its color, and replace it in the set. The cards are thoroughly reshuffled, and I again
choose a card at random, observe its color, and replace it in the set. This is done a total of four times. Let X be
the number of red cards observed in these four trials. The mean of X is
a. 4.
b. 2.
c. 1.
d. 0.5.
5. If X has a binomial distribution with 20 trials and a mean of 5, then the success probability p is
a. 0.25.
b. 0.50.
c. 0.75.
d. not known without taking a sample.
6. If X is B(n = 4, p = 1/4), the variance of X is
a. 1.414.
b. 1.000.
c. 0.866.
d. 0.750.
7. In an instant lottery, your chances of winning are 0.2. If you play the lottery five times and outcomes are
independent, the probability that you win at most once is
a. 0.0819.
b. 0.2.
c. 0.4096.
d. 0.7373.
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82 Chapter 13
8. In an instant lottery, your chances of winning are 0.2. If you play the lottery five times and outcomes are
independent, the probability that you win all five times is
a. 0.6723.
b. 0.3277.
c. 0.04.
d. 0.00032.
Use the following to answer questions 9 – 11.
The superintendent of a large school district reads that 60% of middle school students have a personal site on
myspace.com. She selects a sample of 50 middle school students at random from her district and has them
complete a small survey. One of the questions asks if they have a personal site on myspace.com. Let X denote
the number in the sample that say they have such a site.
9. The mean of X is
a. 20.
b. 30.
10. The standard deviation of X is
a. 3.46.
b. 5.48.
c. 50.
d.60
c. 12.
d. 30.
11. The probability that X is at least 35 is
a. less than 0.0001.
b. about 0.0749
c. about 0.3409.
d. about 0.6591.
Use the following to answer questions 12 – 14.
People with type O-negative blood are universal donors whose blood can safely be given to anyone. Only 7.2%
of the population has O-negative blood. A mobile blood center is visited by 20 donors in the afternoon. Let X
denote the number of universal donors among them.
12. The mean of X is
a. 7.2.
b. 1.44.
13. The standard deviation of X is
a. 1.15.
b. 1.20.
c. 1.34.
d. 0.072.
c. 1.34.
d. 1.44.
14. The probability that X is at least 2 is
a. 0.224.
b. 0.257.
c. 0.427.
d. 0.743.
15. A college basketball player makes 80% of his free throws. At the end of a game, his team is losing by two
points. He is fouled attempting a 3-point shot and is awarded three free throws. Assuming each free throw is
independent, what is the probability that he makes at least two of the free throws?
a. 0.89
b. 0.80
c. 0.64
d. 0.384
Binomial Distributions 83
16. A college basketball player makes 5/6 of his free throws. Assuming free throws are independent, the
probability that he makes exactly three of his next four free throws is
1 3 5 1
a. 4     .
6  6 
1 3 5 1
b.     .
6  6 
1 1 5 3
c. 4     .
6  6 
1 1 5 3
d.     .
6  6 
17. Suppose we roll a fair die 10 times. The probability that an even number occur exactly the same number of
times as an odd number on the 10 rolls is
a. 0.1667.
b. 0.2461.
c. 0.3125.
d. 0.5000.
18. A local politician claims that 1 in 5 automobile accidents involve a teenage driver. He is advocating
increasing the age at which teenagers can drive alone. Over a 2-month period there are 67 accidents in your city,
and only 9 of them involved a teenage driver. If the politician is correct, what is the chance that you would
observe 9 or fewer accidents involving a teenage driver?
a. 0.0524
b. 0.0901
c. 0.1343
d. 0.200
19. A college basketball player makes 80% of his free throws. Over the course of the season, he will attempt
100 free throws. Assuming free throw attempts are independent, what is the probability that he makes at least
90 of these attempts?
a. 0.90
b. 0.72
c. 0.2643
d. 0.0062
Use the following to answer questions 20 – 22.
In a large Metropolitan area, 20% of the families have an adjusted gross income of $80,000 or more reported on
their local income tax return. A random audit chooses 100 of these returns for careful study. Let X be the
number of local income tax returns audited that show an adjusted gross income of under $80,000
20. The mean of X is
a. 7.2.
b. 20.
21. The variance of X is
a. 2.
b. 4.
c. 80.
d. 100.
c. 16.
d. 20.
22. The probability that at least 30 of the returns audited show an adjusted gross income of more than $80,000 is
a. 0.2000.
b. 0.1020.
c. 0.0228.
d. 0.0062.
84 Chapter 13
23. An article in Parenting magazine reported that 60% of Americans needed a vacation after visiting their
families for the holiday. Suppose this is the true proportion of Americans who feel this way. A random sample
of 100 Americans is taken. What is the probability that less than 50% of the people in the sample feel that they
need a vacation after visiting their families for the holidays?
a. 0.4000
b. 0.1446
c. 0.0207
d. 0.0062
24. A simple random sample of 1000 Americans found that 61% were satisfied with the service provided by the
dealer from which they bought their car. A simple random sample of 500 Canadians found the 58% were
satisfied with the service provided by the dealer from which they bought their car. The sampling variability
associated with these statistics is
a. larger for the Canadians because the sample size is smaller.
b. smaller for the sample of Canadians, since the population of Canada is less than half that of the United
States. Hence, the sample is a larger proportion of the population.
c. smaller for the sample of Canadians, since the percentage satisfied was smaller than that for the
Americans.
d. larger for the Canadians, since Canadian citizens are more widely dispersed throughout their country than
American citizens are in the United States. Hence, Canadians have more variable views.
25. The incomes in a certain large population of college teachers have a normal distribution with mean $75,000
and standard deviation $10,000. Four teachers are selected at random from this population to serve on a salary
review committee. What is the probability that their average salary is less than $65,000?
a. .0228
b. 0.1587
c. .9772
d. essentially 0
Use the following to answer questions 26— 27
The average age of cars owned by residents of a small city is 6 years with a standard deviation of 2.2 years. A
simple random sample of 400 cars is to be selected, and the sample mean age x of these cars is to be computed.
26. We know the random variable x has approximately a normal distribution because of
a. the law of large numbers.
b. the central limit theorem.
c. the 68-95-99.7 rule.
d. the fact that probability is the long-run proportion of times an event occurs.
27. The probability that the average age x of the 400 cars is more than 6.1 years is
a. 0.8186.
b. 0.4801.
c. 0.1814.
d. 0.0001.