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Problems
1. If two fair dice are rolled 10 times, what is the probability of at least one 6 (on either die) in exactly
five of these 10 rolls
2. On a random day, the number of vacant rooms of a big hotel in New York City is 35, on average. What
is the probability that next Saturday this hotel has at least 30 vacant rooms
3. Suppose that X is a Poisson random variable with P (X = 1) = P (X = 3). Find the P (X = 5).
4. An absentminded professor does not remember which of his 12 keys will open his office door. If he
tries them at random and with replacement:
a. On average, how many keys should he try before his door opens
b. What is the probability that he opens his office door after only three tries
5. Suppose that 20% of a group of people have hazel eyes. What is the probability that the eighth
passenger boarding a plane is the third one having hazel eyes Assume that passengers boarding the
plane form a randomly chosen group.
6. From a panel of prospective jurors, 12 are selected at random. If there are 200 men and 160 women
on the panel, what is the probability that more than half of the jury selected are women
7. It takes a professor a random time between 20 and 27 minutes to walk from his home to school every
day. If he has class at 9:00 A.M. and he leaves home at 8:37 A.M., find the probability that he reaches
his class on time.
8. Suppose that a Scottish soldier’s chest size is normally distributed with mean 39.8 and standard
deviation 2.05 inches, respectively. What is the probability that of 20 randomly selected Scottish
soldiers, five have a chest of at least 40 inches
9. Suppose that lifetimes of light bulbs produced by a certain company are normal random variables with
mean 1000 hours and standard deviation 100 hours. Is this company correct when it claims that 95%
of its light bulbs last at least 900 hours
10. The lifetime of a TV tube (in years) is an exponential random variable with mean 10. If Jim bought
his TV set 10 years ago, what is the probability that its tube will last another 10 years
11. In a hospital, babies are born at a Poisson rate of 12 per day. What is the probability that it takes at
least seven hours before the next three babies are born
12. The grades of students in a calculus-based probability course are normal with a mean of 72 and standard
deviation 7. If 90, 80, 70, and 60 are the respective lowest, A, B, C, and D, what percent of students
in this course get A’s, B’s, C’s, D’s, and F’s
13. Alan and Susie play a series of backgammon games until one of them wins five games. Suppose that
the games are independent and the probability that Susie wins is 0.58.
a. Find the probability that the series ends in seven games.
b. If the series ends in seven games, what is the probability that Susie wins
1

14. A certain basketball player makes a foul shot with probability 0.45. Determine for what value of k the
probability of k baskets in 10 shots is maximized, and find this maximized probability.
15. A restaurant serves eight fish entrées, 12 beef, and 12 poultry. If customers select from these entrées
randomly, what is the expected number of fish entrées ordered by the next four customers
16. Suppose that a certain bank returns bad checks at a Poisson rate of three per day. What is the
probability that this bank returns at most four bad checks during the next two days
17. When a certain car breaks down, the time that it takes to fix it (in hours) is a random variable with
the density function f(x) = ce −3x if 0 ≤ x < inf and 0 otherwise.
a. Calculate the value of c.
b. Find the probability that when this car breaks down, it takes at most 30 minutes to fix it.
18. The breaking strength of a certain type of yarn produced by a certain vendor is normal with mean 95
and standard deviation 11. What is the probability that, in a random sample of size 10 from the stock
of this vendor, the breaking strength of at least two are over 100
19. If X is the number of 6’s which appear when 72 dice are thrown, what is the expected value of X 2
20. The moment generating function of a random variable X is e 4(et −1) . Show that P (µ − 2σ < X <
µ + 2σ) = 0.931.
21. Accidents occur at an intersection at a rate of three per day. What is the probability that during
January there are exactly three days (not necessarily consecutive) without any accidents
22. The time it takes for a student to finish an aptitude test (in hours) has a density function of the form
f(x) = c(x − 1)(2 − x) if 1 < x < 2 and 0 elsewhere.
a. Determine the constant c.
b. What is the probability that a student will finish the aptitude test in less than 75 minutes Between
1.5 and 2 hours
23. The amount of cereal in a box is normal with mean 16.5 ounces. If the packager is required to fill at
least 90% of the cereal boxes with 16 or more ounces of cereal, what is the largest standard deviation
for the amount of cereal in a box
24. Customers arrive at a restaurant at a Poisson rate of 12 per hour. If the restaurant makes a profit
only after 30 customers have arrived, what is the expected length of time until the restaurant starts to
make a profit
25. The moment generating function of a random variable X is ( 2 3 + 1 3 et ) 9 . Show that
P (µ − 2σ < X < µ + 2σ) =
5∑
i=1
( 9
x)
( 1 3 )x ( 2 3 )9−x
2