Generic Discrete
Find the mean and standard deviation of the following probability distributions. Make a table if one is not
given. Most of you can simply enter the data into your calculator as you did with frequency tables and
have the calculator spit out the answers.
1. When randomly selecting a jail inmate convicted of DWI, the probability distribution for the
number x of prior DWI sentences is as described in the following table.
x
P(x)
0
0.512
1
0.301
2
0.132
3
0.0550
2. If your college hires the next 4 employees without regard to gender, and the pool of applicants is
large with an equal number of men and women, then the probability distribution for the number x
of women hired is described in the following table.
x
P(x)
0
0.0625
1
0.250
2
0.375
3
0.250
4
0.0625
3. To settle a paternity suit, two different people are given blood tests. If x is the number having
group A blood, then x can be 0, 1, or 2 and the corresponding probabilities are 0.36, 0.48, and 0.16
respectively.
4. The number of dinners that typical Americans cook in a week are listed with their probabilities: 0
(0.0800);1 (0.0500); 2 (0.100);3 (0.130); 4(0.150);5 (0.210); 6 (0.0900); 7 (0.190).
5. In a pizza takeout restaurant, the following probability distribution was obtained. The random
variable x represents the number of topping for a large pizza. Find the missing probability and then
the mean and standard deviation.
x
P(x)
0
0.30
1
0.40
2
3
0.06
4
0.04
6. In the following probability distribution, the random variable x represents the number of cars per
household in a town of 1000 households. Find the missing probability and then the mean and
standard deviation.
x
P(x)
0
0.125
1
0.428
2
0.256
3
4
.083
7. In the following probability distribution, the random variable x represents the number of credit
cards that adults have. Find the missing probability and then the mean and standard deviation.
x
P(x)
0
0.49
1
0.05
2
0.32
3
0.07
4
8. The Baltimore Computer House finds that the probabilities of selling 0, 1, and 2 microcomputers
in one day are .562, .298, and .140 respectively.
x
P(x)
0
.562
1
.298
2
.140
BINOMIAL EXPERIMENTS - state S, p, q, n, x, write the probability question, solve, and
calculate the mean and standard deviation.
1. According to Discover magazine, 95% of Indy race drivers will survive a crash under certain
conditions. Given those conditions, what is the probability that exactly 12 of 15 Indy race drivers
will survive a crash?
2. According to the Labor Department, 40% of adult workers have a high school diploma but did not
attend college. If 14 adult workers are randomly selected, find the probability that exactly 12 of
them have a high school diploma but did not attend college.
3. Fifteen percent of sport\compact cars are dark green. Assume that 50 sport\compact cars are
randomly selected. Find the probability that there are exactly 9 dark green cars in such a group of
50.
4. In the binomial table, the probability corresponding to n=3, x=2, and p=.01 is shown as .000. Find
the exact probability represented by this.
5. 53% of those who live in the contiguous 48 states reside within 50 miles of a coastal shoreline. The
Sandi Swimwear Company surveys randomly selected U.S. residents from the 48 contiguous
states. Find the probability that among the first 20 residents selected, exactly 12 live within 50
miles of a coastal shoreline.
6. United Airlines Flight 470 form Denver to St. Louis has an on-time performance of 60%. Find the
probability that among 30 such flights, exactly 20 arrive on time.
7. Thirty percent of college students own VCR’s. The Telektronic Company produced a videotape
and sent copies to 10 randomly selected college students as part of a pilot sales program. Find the
probability that exactly 5 of the 10 college students own VCR’s.
8. In #1, what is the probability that more than 12 survive?
9. In #2, what is the probability that 12 or more have a high school diploma but did not attend
college?
10. In #6, find the probability that among 12 such flights, at least 9 arrive on time.
11. In #7, find the probability that fewer than 5 of the 10 college students own VCR’s. Also find the
probability that at least 5 of the 10 college students own VCR's.
12. A Media General-AP poll showed that 20% of adult Americans are opposed to strict pollution
controls on power plants that burn coal and oil. Assume that an environmental group launches a
campaign to lower that number, and a post- campaign study begins with 15 randomly selected
adults. If the 20% level of opposition hasn’t changed, find the probability that among the 15 adults,
fewer than 3 are opposed to the strict controls.
13. A new IRS employee learns that 70% of all audited taxpayers must pay more money (based on IRS
data). During her first day, she must conduct 8 audits, and she hopes that at least 6 of the audited
taxpayers must pay more money. Find the probability of that happening.
14. Inability to get along with others is the reason cited in 17% of worker firings. Concerned about her
company’s working conditions, the personnel manager at the Flint Fabric Company plans to
investigate the 5 employee firings that occurred over the past year. Assuming that the 17% rate
applies, find the probability that among those 5 employees, the number fired because of an
inability to get along with other is at least 3.
15. In a study of stereotypes on infant, 77% of female babies choose a baby doll over a football. If 18
female babies are chosen, what is the probability that exactly 10 of them will choose a baby doll.
16. In the previous problem, what is the probability that at more than 16 of them will choose a baby
doll?
17. A recent survey claims that 87% of men named Bubba own a shotgun. If 9 Bubba's are randomly
chosen, what is the probability that exactly 6 of them will own a shotgun?
18. In the previous problem, what is the probability that at least 8 of them will own a shotgun?
19. At the College of Charleston, 40% of the students have meal plans. If 11 College of Charleston
students are randomly selected, what is the probability that more than 7 of them have meal plans?
20. a) If you flip a perfectly fair coin 10 times, what is the probability of getting exactly 5 heads? b) If
you flip a perfectly fair coin 10 times, what is the probability of getting more than 5 heads? c) If
you flip the coin 100 times, what is the probability of getting exactly 50 heads? d) If you flip the
coin 100 times, what is the probability of getting exactly 49 heads?
Normal Probabilities - Draw a normal graph for each problem. Include x scores(except for #1 and
#2) and z scores and shade and/or label appropriate areas.
1. If the random variable z has a mean of 0 and a standard deviation of 1, find the following
probabilities a)P(z < -1.34), b)P(z < 2.10), c)P(z < 0.00), d)P(z > -1.645), e)P(z > 2.54),
f)P(z > .80), g)P(-2.87 < z < -1.14),h)P(-2.60 < z < .74), i)P(-1.79 < z < 0.00), j)P(.82 < z < 2.28)
2. Find the following areas (or probabilities) or the nearest area in the standard normal table and state
the corresponding z score. Draw and label a normal graph for each. : a).0091 b).1492 c).9842
d).5987 e).2709 f).2514 g).7486 h).8997 i).75 j).25 k).8 l).9 m).1 n).2 o).95 p).05 q).005 r).995
3. Replacement times for CD players are normally distributed with a mean of 7.1 years and a
standard deviation of 1.4 years. a)Find the probability that a randomly selected CD player will
have a replacement time less than 8.0 years. b)What is the percentage of CD players that will have
replacement times less than 8.0 years? c)If 50 CD players are randomly selected, how many of
these would you expect to have replacement times less than 8.0 years? d) Find the probability that
a randomly selected CD player will have a replacement time more than 8.0 years. e)Find the
probability that a randomly selected CD player will have a replacement time between 6.5 and 8.0
years.
4. Assume that the weights of paper discarded by households each week are normally distributed
with a mean of 9.4 lb and a standard deviation of 4.2 lb. a)Find the probability of randomly
selecting a household and getting one that discards between 5.0 lb and 8.0 lb of paper in a week.
b)What is the percentage of households that discard between 5.0 lb and 8.0 lb of paper in a week?
c)In a random selection of 75 households, how many of them do you expect to discard between 5.0
lb and 8.0 lb of paper in a week?
5. One classic use of the normal distribution is inspired by a letter to Dear Abby in which a wife
claimed to have given birth 308 days after a brief visit from her husband, who was serving in the
Navy. The lengths of pregnancies are normally distributed with a mean of 268 days and a standard
deviation of 15 days. a)Given this information, find the probability of a pregnancy lasting 308
days or longer. b)What is the percentage of pregnancies that last 308 days or longer? c)In a
random selection of 5000 pregnancies, how many of them do you expect to last 308 days or longer.
6. Men spend an average of 11.4 min in the shower. Assume that the times are normally distributed
with a standard deviation of 1.8 min. a)If a man is randomly selected, find the probability that he
spends at least 10.0 min in the shower. b)What percentage of men spend at least 10.0 min in the
shower? c)In a random selection of 150 men, how many of them do you expect to spend at least
10.0 min in the shower?
7. According to the International Mass Retail Association, girls aged 13 to 17 spend an average of
$31.20 on shopping trips in a month. Assume that the amounts are normally distributed with
standard deviation of $8.27. a)If a girl in that age category is randomly selected, what is the
probability that she spends between $34.00 and $40.00 in one month? b)What percentage of girls
this age spend between $34.00 and $40.00 in one month? c)If 90 girls in this age group are
randomly selected, how many of them do you expect to spend between $34.00 and $40.00 in one
month?
8. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Mensa is an
organization for people with high IQs, and eligibility requires an IQ above 131.5. a)If a person is
randomly selected, what is the probability that he or she will be eligible for Mensa? b)What
percentage of people meet the Mensa requirement? c)In a typical region of 75,000 people, how
many are eligible for Mensa?
9. Assume that women's heights are normally distributed with a mean of 63.6 in and a standard
deviation of 2.5 in. a)If a woman is randomly selected, what is the probability that she will have a
height greater than 58.1 in? b)What percentage of women have a height greater than 58.1 in?
c)In a random selection of 9000 women, how many would you expect to have a height greater than
58.1 in?
10. Some vending machines are designed so that their owners can adjust the weights of the quarters
that are accepted. If many counterfeit coins are found, adjustments are made to reject more coins,
with the effect that most of the counterfeit coins are rejected along with many legal coins. Assume
that quarters have weights that are normally distributed with a mean of 5.67 g and a standard
deviation of .07 g. a)If a vending machine is adjusted to reject quarters weighing less than 5.50 g
or more than 5.80 g, what is the percentage of legal quarters that are rejected? b)If 800 legal
quarters are inserted into the vending machine, how many of them do you anticipate will be
rejected?
11. Prison terms of convicted embezzlers are normally distributed with a mean of 22.1 months and a
standard deviation of 4.2 months. a)What percentage of convicted embezzlers have a prison term
more than 24 months? b)In a random selection of 80 convicted embezzlers, how many of them do
you expect to have a prison term more than 24 months?
12. In the previous problem, what percentage of convicted embezzlers have a prison term less than
15 months?
13. The nicotine content in Camel cigarettes is normally distributed with a mean of 45.6 mg and a
standard deviation of 3.4 mg. a)In a randomly selected Camel cigarette, what is the probability
that the nicotine content is between 38 mg and 42 mg? b)In a pack of 20 Camel cigarettes, how
many of them do you expect to have a nicotine content between 38 mg and 42 mg?
14. In the previous problem, what is the probability that the nicotine content is greater than 47 mg
or less than 40 mg?
Normal Scores
15. In #3, find the replacement time separating the top 45% from the bottom 55%.
16. In #4, find the weight that separates the bottom 33% from the top 67%.
17. In #6, find the values of the quartiles Q1 and Q3.
18. In #9, find the values of P85, P66, P15.
19. The scores on a test were normally distributed with a mean of 484 and a standard deviation of 52.
Find the values of P21, P50, Q3, and D9.