Problem Set Section 4.1
Discrete Random Variables
1. State whether the variable is discrete or continuous.
a. The number of cups of coffee sold in a cafeteria during lunch.
b. The height of a basketball player.
c. The cost of a Statistics textbook.
d. The blood pressure of a group of students before a Statistics test.
e. The number of goals scored in a soccer game.
2. Write the following in symbolic form using x as the random variable. These are
statements for discrete random variables.
a. exactly four
b. more than four
c. fewer than four
d. at least four
e. at most four
3. Determine whether the distribution represents a probability distribution. If not, identify
the requirement that is not satisfied.
4. In a recent little league softball game, each player went to bat 4 times. The number of
hits made by each player is described by the following probability distribution.
Number of hits, x
0
1
2
3
4
Probability, P(x)
0.10
0.20
0.30
0.25
0.05
a. Verify a probability distribution is given?
b. What is the mean of the probability distribution?
c. What is the standard deviation of the probability distribution?
5. The number of adults living in homes on a randomly selected city block is described by
the following probability distribution.
Number of adults, x
1
2
3
4
Probability, P(x)
0.25
0.50
0.15
0.10
a. Verify a probability distribution is given?
b. What is the mean of the probability distribution?
c. What is the standard deviation of the probability distribution?
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6. The number of credit cards that adults have is described by the following probability
distribution.
Number of credit cards, x
0
1
2
3 or more
Probability, P(x)
0.49
0.32
0.04
???
a. What is the probability that a randomly selected adult has 3 or more credit
cards?
b. What is the probability that a randomly selected adult has at least 2 credit cards?
c. What is the probability that a randomly selected adult has at least 1 credit card?
Find this probability using two different methods.
d. Why can’t you find the mean and standard deviation for this probability
distribution?
7. Number games in a Baseball World Series: The baseball world series a best of 7 game
series. There is a .1809 probability it will last four games, a .2234 chance it will last five
games, and a .2234 chance it will last six games.
a. What is the probability it will last 7 games?
b. Find the mean and standard deviation.
c. Is it unlikely for the World Series to be a sweep by winning in four games?
Explain.
d. Is it unusual for the World Series to be a sweep by winning in four games?
Explain.
8. Expected value of a Winning a TV: One thousand tickets are sold at $1 each. One ticket
will be randomly selected and the winner will receive a TV valued at $350. What is the
expected value if a person buys one ticket?
9. Expected Value in Craps: When you bet $5 on the pass line in craps there is a 244/495
probability that you will win $5. What is your expected value? In the long run how
much would you expected to lose for each dollar bet?
10. Expected Value in Roulette: When you bet $5 on the number 3 and the number comes
up you win $175. If you bet $5 that the number is even you would win $5.
a. What is the probability of winning if you bet a 3 will come up? What is the
expected value?
b. What is the probability of winning if you bet an even number will come up?
What is the expected value?
c. In the long run, which is the better bet?
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11. Expected Value of Dice Game: In this game you roll one die. You bet $6 that you roll a
3. If the game is fair your total profit(winnings) would be $30. In the long run, what
would you expected to win playing this game? Set up a table to compute the expected
value.
12. Expected Value of a Life Insurance Policy: A twenty-five–year-old man decided to pay
$600 for a one-year life insurance with coverage for $1,000,000. The probability of him
living through the year is .9995.
a. From the perspective of the man, what is the expected value?
b. In the long run, what is the insurance company’s expected profit per policy?
c. If the insurance company just wanted to break even instead of making a profit,
what would the cost of the policy be?
13. Expected Value of Extended Warranties: Best Buy was offering an extended warranty
on a $6000 home entertainment system. The cost of the warranty is $700. The
probability that Best Buy would have to replace this system is 1 in a 1000. In the long
run, what is Best Buy’s expected profit from each home entertainment system warranty
sold.
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Problem Set Section 4.2
Binomial Probability Distribution
1. Determine whether the experiment is a binomial experiment. If it is not, explain why?
Let X be the random variable.
a. You observe the gender of the next 100 babies born at a local hospital.
X=number of girls.
b. You roll a die 100 times. X=the number that appears.
c. You spin a number wheel that has 10 numbers 100 times. X=winning numbers
on each spin of the wheel.
d. You test four pain relievers. X=pain reliever that is most effective.
e. Test a pain reliever using 20 people to determine if it is effective. X=the number
of people who find the pain reliever effective.
f. A man plays a game in which he has a 30% chance of winning. X=number of
times he wins in 10 weeks.
2. Find the following binomial probabilities. Write out the probability statement using the
notation P(X=x).
a. x = 5 ; n = 10 ; p = .5
b. x = 2 ; n = 8 ; p = .2
c. x = 0 ; n = 10 ; p = .9
d. x = 4 ; n = 15 ; p = .25
3. If 1/4 of the roses grown are white, find the probability that 2 out of 5 will be white.
4. If 20% of all automobiles in a large city are foreign-made, find the probability that if
there are eight automobiles in a parking lot, three are foreign-made.
5. Probability of Gender at Birth: Assume that male and female births are equally likely
and independent.
a. Find the probability of 4 boys in 10 births.
b. Find the probability of at most 4 boys in 10 births.
c. Find the probability of 5 boys and 5 girls in 10 births. Is this unlikely?
d. At what point does the probability of the number of boys in 10 births become
unlikely?
6. Free Throw Shots: Suppose that each time you take a free throw shot, you have a 25%
chance of making it. If you take 15 shots,
a. What is the probability of making exactly 5 of them?
b. What is the probability of making fewer than 3 shots?
c. What is the probability of making no shots?
d. What is the probability of making at least one shot?
e. At what point does the number of shots made become unlikely?
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7. True / False Test: A student takes an eight question true-false exam and guesses on
each question. Find the probability of passing if the lowest passing grade is six out of
eight. That is they must get 6, 7 or 8 correct.
8. Multiple Choice Test: A Multiple Choice test has four possible answers to each of 16
questions. A student guesses the answer to each question.
a. What is the probability of guessing one answer correct?
b. What is the probability of 8 wrong and 8 correct?
c. If you have to get at least 12 correct answers to pass, what is the probability of
passing this test? Would it be unlikely to guess answers and still pass?
9. Multiple Choice Test: If a Multiple Choice Test has five choices for each question, find
the probability of guessing at least six out of eight correctly. Would it be unlikely to
guess answers and still pass if you must get at least 6 answers correct?
10. Wearing Glasses: The proportion of juvenile delinquents who wear glasses is known to
be 0.2 whereas the proportion of non-delinquents wearing glasses is 0.6. A researcher
plans to obtain random samples of 15 delinquents and 20 non-delinquents. Assume that
the populations of delinquents and non-delinquents are very large. Is it unlikely for 2
delinquents to wear glasses? How about 2 non-delinquents?
11. Heart Transplants: The proportion of patients in a large medical database who survived
a heart transplant operation is 0.8. Assume that randomly sampling 20 heart-transplant
patients from this database satisfies (at least approximately) the conditions of a
binomial experiment. What is the probability that no one will die (all live)?
12. Cold Cures: Historical information is available on two cold drugs: A and B. The
probability Drug A cures a randomly selected patient is 0.40, whereas the probability for
Drug B is 0.60. A physician treats 20 randomly selected patients with Drug A and 20
patients with Drug B. The conditions of the binomial distribution are assumed to hold.
What is the probability that 10 patients will be cured if they took Drug A? Drug B?
13. Gecko Gender: For the leopard gecko (Eublepharis macularius), the gender of their
offspring is determined by the temperature during embryonic development. At 30
degrees C, the offspring is almost completely female. At 32.5 degrees C the offspring is
almost completely male. Researchers at the University of Texas have determined that
at 31 degrees C, the proportion of males produced is 35%. They have 10 leopard gecko
embryos that have been incubated at 31 degrees C and will soon hatch. Suppose they
are interested in the number of male leopard geckos from this litter of 10. Would it be
unlikely for 8 of the 10 embryos to be male?
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14. Emergency Room Use: If 20% of the people in a community use the emergency room at
a hospital in one year, find these probabilities for a sample of 7 people.
a. At most three used the emergency room.
b. Exactly three used the emergency room.
c. At least five used the emergency room.
15. German Measles: If 60% of the children had German measles by the time they were 12
years old, find these probabilities for a sample of 8 children.
a. At least 5 have had German measles.
b. Exactly 7 have had German measles.
c. More than 3 have had German measles.
16. Wearing Glasses: A recent study found that 80% of all people over 60 years of age wear
glasses. If a random sample of 6 people over 60 years of age is selected, find these
probabilities.
a. Exactly 4 people will wear glasses.
b. At most 2 people will wear glasses.
c. At least 2 people will wear glasses.
17. Pizza Delivery: There is a 90% chance that Square Table Pizza will make a delivery in
less than 30 minutes. An executive for Square Table decides to test the delivery system
by ordering 20 pizzas at different times and different locations.
a. Find the probability that all 20 pizzas will be delivered in less that 30 minutes.
b. Find the probability that all 20 pizzas will be delivered in 30 minutes or more. Be
careful. What is “p” in this case?
c. Find the probability that more than one pizza will be delivered in 30 minutes or
more.
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Problem Set Section 4.3
Mean, Variance and Standard Deviation
for the Binomial Probability Distribution
1. Find the mean, variance, and standard deviation for each of the values of n and p when
the conditions for the binomial distribution are met.
a. n = 100, p = 0.75
b. n = 20, p = 0.5
c. n = 1000, p = 0.1
d. n = 50, p = 2/5
e. n = 36, p = 1/6
2. Find the mean, variance, and standard deviation for the number of heads when ten
coins are tossed. Would it be unusual to get 7 heads? Explain using 2σ rule.
3. A student takes an eight question true-false exam and guesses on each question. Find
the mean, variance and standard deviation. If the lowest passing grade is six out of
eight. Would it be unusual to pass? Explain using 2σ rule.
4. Tasteful Tomato Seeds have an 80% germination rate. Find the mean, variance, and
standard deviation of a package of 200 seeds. Would it be unusual for 150 seeds to
germinate? Explain using 2σ rule. Would the probability for 150 seeds to germinate be
unlikely? Explain using the 0.05 rule. What is your observation?
5. According to government data, the probability that an adult was never married is 15%.
In a random survey of 10 adults, what is the mean and standard deviation of the
number that never married? Would it be unusual for 5 of the 10 to have never married?
Explain using 2σ rule.
6. The probability that a house in an urban area will be burglarized is 5%, if 20 houses area
randomly selected, what is the mean of number of houses burglarized? What are the
minimum and maximum unusual values for the number of houses burglarized?
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Problem Set Section 4.4
Other Discrete Probability Distributions
1. Poisson: On an average Friday, a waitress gets no tip from 5 customers. Find the
probability that she will get no tip from 7 customers this Friday.
2. Poisson: During a typical football game, a coach can expect 3.2 injuries. Find the
probability that the team will have at most 1 injury in this game.
3. Poisson: It has been observed that the average number of traffic accidents requiring
medical assistance on the Hollywood Freeway between 7 and 8 PM on Wednesday
mornings is 1. What, then, is the chance that there will be a need for exactly 2
ambulances on the Freeway, during that time slot on any given Wednesday morning?
The hospital dispatcher needs to know.
4. Poisson: It is tent inspection time on the Provo Proving Ground, and any irregularities
will be cause for extra KP duty. The tent manufacturer's established track record is 1
weaving fault per 50 square feet of canvas. If Barney's tent contains 75 square feet of
canvas, and he is otherwise an immaculate tent keeper, what are his chances of pulling
extra KP?
5. Poisson: The switchboard in a Denver law office gets an average of 2.5 incoming phone
calls during the noon hour on Thursdays. Experience shows that the existing lunch hour
staff can handle up to 5 calls in an hour. They seem to be well covered. But just to test
the edges, what is the actual chance that 6 calls will be received during the lunch period,
on some particular Thursday?
6. Poisson: The average number of bad checks received at Exchange Bank of Mariposa is 6
per day. Every morning, upbeat manager Zena bets herself that there will be only 4 bad
checks on that day. If she keeps this up for a month, how many times in that month will
she probably win her bet?
7. Poisson: Idaho's Hidden Lake is continually stocked with trout, to a point where guest
anglers average 2 trout per hour of fishing. Ardent suitor Sam brings avid trouter Muriel
to Hidden Lake Lodge, promising her a rewarding afternoon of sport. Muriel has little
patience with incompetence, and will return the ring if in 2 hours she catches nothing.
What are the odds that the marriage will take place?
8. Geometric: A couple decides that they will have kids until a girl is born. The outcome
of each birth is independent event, and the probability that a girl will be born is 1/2.
The birth at which the first girl appears is a geometric distribution. What is the expected
family size?
9. Geometric: Jason is rolling a die. Calculate the probability of getting a 1 on the 5th roll.
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10. Geometric: A top NHL hockey player scores on 93% of his shots in a shooting
competition. What is the probability that the player will not miss the goal until his 20th
try?
11. Negative Binomial: Suppose we are at a rifle range with an old gun that misfires 5 out
of 6 times. Define ``success'' as the event the gun fires and let X be the number of
failures before the third success. Find the probability that there are 10 failures before
the third success?
12. Negative Binomial: Bob is a high school basketball player. He is a 70% free throw
shooter. That means his probability of making a free throw is 0.70. During the season,
what is the probability that Bob makes his third free throw on his fifth shot?
13. Hypergeometric: Suppose you select randomly select 12 cards without replacement
from an ordinary deck of playing cards. What is the probability that EXACTLY 7 of those
cards will be black (i.e., either a club or spade)?
14. Hypergeometric: Suppose we randomly select 5 cards without replacement from an
ordinary deck of playing cards. What is the probability of getting exactly 2 red cards (i.e.,
hearts or diamonds)?
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Selected Answers:
Section 4.1
1a) discrete
2a) P(x=4)
3a) yes
5b) 2.1
6a) 0.15
8) -0.65
9a)-$0.07
10a) -0.26
11) nothing
12a) -100
13) 694
Section 4.2
1a) yes
3) 0.264
10) no, yes
1b) continuous
2b) P(x>4)
3b) no
5c) 0.9
6b) 0.19
1c) continuous
2c) P(x<4)
3c) yes
1d) continuous
2d) P(x>4)
3d) no
1e) discrete
2e) P(x<4)
1d) no
8a) 0.0535
1e) yes
8b) 0.197
6c) 0.51
9b) 1.4¢
10b) -0.26
10c) neither
12b) 100
12c) $500
1b) no
5a) 0.205
1c) n0
5b) 0.377
Section 4.3
2) µ=5 σ2=2.5 σ =1.6 no
4) µ=160 σ2=32 σ =5.7 , 2 no, 0.05 yes
Section 4.4
13. 0.209662799 14. [26C2 ] [ 26C3 ] / [ 52C5 ]
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1f) yes
8c) 1 – P(x≤11)