Download Probability - Mr. Taylor`s Math

AP Statistics
Probability - Major Topic III
2001 Free Response
A department supervisor is considering purchasing one of two comparable photocopy machines, A or B.
Machine A costs $10,000 and machine B costs $10,500. This department replaces photocopy machines
every three years. The repair contract for machine A costs $50 per month and covers an unlimited
number of repairs. The repair contact for machine B costs $200 per repair. Based on past performance,
the distribution of the number of repairs needed over any one-year period for machine B is shown below.
Number of Repairs
0
1
2
3
Probability
0.50 0.25 0.15 0.10
You are asked to give a recommendation based on overall cost as to which machine, A or B, along with
its repair contact, should be purchased. What would your recommendation be? Give a statistical
justification to support your recommendation.
2005 Free Response
Let the random variable X represent the number of telephone lines in use by the technical support center
of a software manufacturer at noon each day. The probability distribution of X is shown in the table
below.
x
0
1
2
3
4
5
p( x) 0.35 0.20 0.15 0.15 0.10 0.05
(a) Calculate the expected value (the mean) of X.
(b) Using past records, the staff at the technical support center randomly selected 20 days and found that
an average of 1.25 telephone lines were in use at noon on those days. The staff proposes to select
another random sample of 1,000 days and compute the average number of telephone lines that were in
use at noon on those days. How do you expect the average from this new sample to compare to that
of the first sample? Justify your response.
(c) The median of a random variable is defined as any value x such that P(X < x) > 0.5 and P(X > x) >
0.5. For the probability distribution shown in the table above, determine the median of X.
(d) In a sentence of two, comment on the relationship between the mean and the median relative to the
shape of this distribution.
Made
shot
Missed
Shot
Total
Males
Females
Total
a) What is the probability that a shot was made?
b) What is the probability that a shot was taken by a female?
c) What is the probability a shot was made and the shooter was a female?
d) What is the probability a shot was made or the shooter was a female?
e) Given that a shot was made, what is the probability the shooter was a male?
f) Given that the shooter was female, what is the probability the shot was missed?
g) Assuming the conditions are met, what is the probability that of 5 randomly selected males, 2 missed
their shot?
h) Assuming the conditions are met, what is the probability that of 10 randomly selected females, 5 made
their shot?
i) Assuming the conditions are met, what is the probability that it takes 6 randomly selected males until a
shot is missed?
j) Assuming the conditions are met, what is the probability that the first miss is on the 4th shot?
k) Assuming the conditions are met, if 50 shots were taken, how many missed shots would we expect?
The standard deviation?
l) Are making the shot and being male independent? Justify your answer.
Microcomputers are shipped to the University bookstore from three factories A, B, and C. You know that
factory A produces 20% defective microcomputers, whereas B produces 10% defectives and C only 5%
defectives. The manager in the store receives a new shipment of microcomputers and discovers that 40%
are from factory C, 40% are from factory B, and 20% are from factory A. (Hint: make a tree diagram)
a) What is the probabilities of finding a defective microcomputer in this shipment?
b) Are the events “microcomputer comes from factory A” and “microcomputer comes from
factory B” mutually exclusive? Are they independent?
c) Suppose the manager randomly selects one microcomputer, and discovers that it is defective.
What is the probability that it came from factory A?
2002 Free Response
Airlines routinely overbook flights because they expect a certain number of no-shows. An airline runs a 5
p.m. commuter flight from Washington, D.C., to New York City on a plane that holds 38 passengers.
Past experience has shown that if 41 tickets are sold for the flight, then the probability distribution for the
number who actually show up for the flight is as shown in the table below.
Number who
actually show up
Probability
36
37
38
39
40
41
0.46
0.30
0.16
0.05
0.02
0.01
Assume that 41 tickets are sold for each flight.
a) There are 38 passenger seats on the flight. What is the probability that all passengers who show up
for this flight will get a seat?
b) What is the expected number of no-shows for this flight?
c) Given that not all passenger seats are filled on a flight, what is the probability that only 36 passengers
showed up for the flight?
2003 Free Response
Contestants on a game show spin a wheel like the one shown in the figure above. Each of the four
outcomes on this wheel is equally likely and outcomes are independent from one spin to the next.

The contestant spins the wheel.

If the result is a skunk, no money is won and the contestant’s turn is finished.

If the result is a number, the corresponding amount in dollars is won. The contestant can then stop
with those winnings or can choose to spin again, and his or her turn continues.

If the contestant spins again and the result is a skunk, all of the money earned on that turn is lost
and the turn ends.

The contestant may continue adding to his or her winnings until he or she chooses to stop or until
a spin results in a skunk.
(a)
(b)
What is the probability that the result will be a number on all of the first three spins of the wheel?
Suppose a contestant has earned $800 on his or her first three spins and chooses to spin the wheel
again. What is the expected value of his or her total winnings for the four spins?
2006 Free Response Question (modified)
Golf balls must meet a set of five standards in order to be used in professional tournaments. One of these
standards is distance traveled. When a ball is hit by a mechanical device, Iron Byron, with a 10-degree
angle of launch, a backspin of 42 revolutions per second, and a ball velocity of 235 feet per second, the
distance the ball travels may not exceed 291.2 yards. Manufacturers want to develop balls that will travel
as close to the 291.2 yards as possible without exceeding that distance. A particular manufacturer has
determined that the distances traveled for the balls it produces are normally distributed with a standard
deviation of 2.8 yards. This manufacturer has a new process that allows it to set the mean distance the
ball will travel.
(a) If the manufacturer sets the mean distance traveled to be equal to 288 yards, what is the
probability that a ball that is randomly selected for testing will travel too far?
(b) Assume the mean distance traveled is 288 yards and that balls are independently tested. What is
the probability that the first ball traveling too far will not occur until the fourth ball?
(c) Assume the mean distance traveled is 288 yards and that five balls are independently tested. What
is the probability that three of the five balls will exceed the maximum distance of 291.2 yards?
(d) Assume the mean distance traveled is 288 yards and that five balls are independently tested. What
is the probability that six balls in a row will exceed the maximum distance of 291.2 yards?
(e) If the manufacturer wants to be 99 percent certain that a randomly selected ball will not exceed the
maximum distance of 291.2 yards, what is the largest mean that can be used in the manufacturing
process?
2002 Free Response Question (modified)
There are 4 runners on the New High School team. The team is planning to participate in a race in which
each runner runs a mile. The team time is the sum of the individual times for the 4 runners. Assume that
the individual times of the 4 runners are all independent of each other. The individual times, in minutes,
of the runners in similar races are approximately normally distributed with the following means and
standard deviations.
Runner 1
Runner 2
Runner 3
Runner 4
Mean
4.9
4.7
4.5
4.8
Standard Deviation
0.15
0.16
0.14
0.15
(a) Runner 3 thinks that he can run a mile in less than 4.2 minutes in the next race. Is this likely to
happen? Explain.
(b) Runner 3 believes the probability estimated in part (a) is much too low. He plans on running until
he gets 4.2 minutes three times in a row. Explain what probability setting is implied and whether
the probability calculated will be valid.
(c) The distribution of possible team times is approximately normal. What are the mean and standard
deviation of this distribution?
(d) Suppose the team’s best time to date is 18.4 minutes. What is the probability that the team will
beat its own best time in the next race?
(e) Knowing there is a track meet every weekend, what is the probability that the team will beat its
own best time by the time they have raced twice?