1. Ilia and Neil want to kill some time while their students are taking the final exam, and
they decide to play a game of chance. Ilia claims he can obtain at least one 5 in six
rolls of a five-sided die (on which the numbers 1 through 5 occur with equal
probability). Neil claims he can obtain at least one Heads in four flips of a fair coin.
a. Is this a fair bet? That is, do Ilia and Neil have the same probability of
achieving their respective claims?
b. Ayse decided to join the game. She claims that she can obtain at least 15
Heads in 30 flips of a fair coin. What are her chances?
c.
The three professors are in Antonio's office. While playing, they observe that
Antonio's secretary receives on average 2 telephone calls per ten minutes. She
orders the professors to answer the calls during her half-hour coffee break,
and before she leaves she says, "Don't worry. I have already received 10 calls
in the first hour of the morning. You won't get a lot of calls." What is the
probability that the professors receive at least 5 calls during the coffee-break?
What assumptions are you making?
1. Michèle Hibon works as a tutor for the Applied Statistics course two hours per day,
four days a week. On average, about 6 students per hour come to see her, and the
arrivals appear to be independent and stationary (occur at a stable rate). Twenty
minutes after the tutoring hours begin one day, Michèle wants to take a coffee break
for 15 minutes but doesn’t want to miss any students who might come to see her. So,
she wonders what are the chances than no students will arrive during the next 15
minutes. Find this probability
(a) if two students arrived in the past ten minutes,
(b) if no students have come in to see her yet on this particular day.
2. On the basis of past data regarding sales, the owner of a car dealership finds that on
average 3.75 cars are sold per day on Saturdays and Sundays during the months of
January and February. The dealership is open for ten hours on Saturdays and
Sundays. The sales rate is relatively stable for different hours of the day and
purchases appear to be independent of one another. The owner has no reason to
believe that this year’s carselling process will be different from that in the past years
covered by the data. On Saturday, February 4, the dealership will open at 9am. What
is the probability that the first sale of the day will occur before 11am?
3. The number of incoming calls to a telephone switchboard can be modeled
successfully as a Poisson random variable (this has been shown in several studies).
Suppose that on average the switchboard receives 3 calls per minute.
a) Find the probability that at most 3 calls are received in a minute.
b) What is the probability that no calls are received in the next 30 seconds?
4. The number of calls to a toll-free number for P&G customer service between 8
a.m. and 5 p.m., a nine-hour day, averages 240 per day. The load is relatively
stable throughout the day and the calls appear to be independent. For the past
several years, approximately 30% of the calls each day pertain to the moon-andstars logo used by P&G and whether or not P&G is devoted to satanic worship.
These calls also occur independently at a stable rate throughout the day. The CEO
of P&G personally wants to hear some of these calls concerning satanic worship.
The CEO decides to intercept all incoming calls for a 15-minute period on a given
day. What is the probability that the CEO will encounter at least one caller asking
about P&G’s devotion to satanic worship?
5. The time it takes a student to go from her house to school is normally distributed
with a mean of 20 minutes and a standard deviation of 5. Estimate the percentage
of time she will be late for class if she leaves her house 30 minutes before the start
of class. If you were her, how much time would you allow to go to school each
day?
6. An architect designing the men’s gymnasium at a university wants to make the
interior doors high enough, so that 95% of the men will have at least a 1-foot
clearance. Assuming that the heights will be normally distributed, with a mean of 70
inches and a standard deviation of 3 inches, how high must the architect make the
doors?
(a) A spacecraft has ten rocket engines, at least nine of which must function properly on
takeoff for the craft to escape the atmosphere and avoid crashing back to earth. Scientists
have determined that each rocket has a 95% chance of working correctly on takeoff. The
probabilities are independent. What is the probability that the spacecraft will take off
successfully?
(b) A competing spacecraft has 100 smaller rockets, at least 90 of which must function
properly on takeoff to avoid falling back to Earth. Each of these rockets has a 95%
chance of working correctly on takeoff. The probabilities are independent. What is the
probability that the spacecraft will take off successfully? How does this compare with
your answer to part (a)?
8 An important quality characteristic for soft-drink bottlers is the amount of soft drink in
the bottle (just think what Pepsi could do if Coke bottles were sometimes only half full).
In a particular filling process (when working as usual), the number of ounces injected
into a 12-ounce bottle is normally distributed with a mean of 12.00 ounces and a standard
deviation of 0.04 ounces. Bottles that contain less than 11.90 ounces do not meet the
bottler’s quality standard and are sold at a substantial discount.
(a) What is the probability that a randomly selected bottle will fail to meet the quality
standard?
(b) When the filling system’s compressor fails, the distribution of the system shifts to a
normal distribution with a mean of 11.95 ounces and a standard deviation of 0.2 ounces.
What is now the probability that a randomly selected bottle will fail to meet the quality
standard? If 20,000 bottles are filled with this faulty process, what is the probability that
more than 8,000 of them will fail to meet the quality standard?