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Name____________________________
AP Statistics @ Harrison High
Random Variables Test
Important Formulas:
If X is a random variable and a and b are fixed numbers, then
If X and Y are random variables, then
a bX  a  b X
 X Y   X  Y
If X is a random variable and a and b are fixed numbers, then
 a2bX  b 2 X2
If X and Y are independent random variables, then . . .
 X2 Y   X2   Y2
 X2 Y   X2   Y2
If X and Y have correlation p, then . . .
 X2 Y   X2   Y2  2 p X  Y
 X2 Y   X2   Y2  2 p X  Y
For binomials . . .
n
P( X  k )    p k (1  p)n k
k 
  np
  np(1  p)
For geometrics . . .
P( X  n)  (1  p)n1 p

1
p
P( X  n)  (1  p)n

(1  p )
p2
Part I - Multiple Choice (Questions 20-31) - Circle the answer of your choice.
20. Which of the following are true statements?
I.
II.
III.
(a)
(b)
(c)
(d)
(e)
The histogram of a binomial distribution with p = .5 is always symmetric.
The histogram of a binomial distribution with p = .9 is skewed to the right.
The histogram of a geometric distribution with p = .3 is always skewed right.
I and II
I and III
II and III
I, II, and III
None of the above gives the complete set of complete responses.
21. Binomial and geometric probability situations share many conditions. Identify the choice
that is not shared.
(a)
(b)
(c)
(d)
(e)
The probability of success on each trial is the same.
There are only two outcomes on each trial.
The random variable is the number of successes in a given number of trials.
The probability of a success equals 1 minus the probability of a failure.
The mean depends on the probability of a success.
22. 3,600,000 dice are rolled. Determine the probability that between 599,000 and
610,000 4’s appear (think Normal approximation to the binomial!).
(a)
(b)
(c)
(d)
(e)
0.67
0.74
0.92
0.08
ERR:DOMAIN
23. A probability experiment involves a series of identical, independent trials with two
outcomes (success/failure) per trial and the probability of a success on each trial is 0.1.
Determine the number of trials, n, in a binomial experiment such that the expected number
of successes in that binomial experiment will be equal to the expected number of trials in a
geometric experiment.
(a)
(b)
(c)
(d)
(e)
2
5
10
50
100
24. In which of the following games would you have the best chance of winning?
(a)
(b)
(c)
(d)
(e)
Toss a coin 20 times. You win if you get more than 11 heads.
Toss a coin 10 times. You win if you don’t get 4, 5, or 6 heads.
Toss a coin 7 times. You win if you get at least 5 heads.
Toss a coin 4 times. You win if you get at least 3 heads.
Toss a coin 5 times. You win if you get exactly 3 heads.
25. A brand of model rocket kit, when finished, produces rockets with a mean mass of 134
grams and a standard deviation of 4.0 grams. The model rocket engine required to fly the
rocket is purchased separately and has a mass with mean 36 grams and standard deviation
2.0 grams. If a randomly selected engine is inserted into a randomly selected rocket, what
are the mean and standard deviation of the total mass?
(a) mean = 85 grams, standard deviation = 3.0 grams
(b) mean = 85 grams, standard deviation = 3.2 grams
(c) mean = 170 grams, standard deviation = 4.5 grams
(d) mean = 170 grams, standard deviation = 6.0 grams
(e) The mean = 170 grams, but the standard deviation cannot be determined from the
information given.
26. A high school golf team of five players is to be in an upcoming tournament. Each of the
players on the team will play a round of golf and the team score is the sum of the five
individual scores. The individual player scores are independent of each other and
approximately normally distributed with the following means and standard deviations.
Golfer
Mean
Standard Deviation
1
78
3
2
79
4
3
81
2
4
84
4
5
93
6
What are the mean and standard deviation of the team score?
(a)
(b)
(c)
(d)
(e)
Mean = 83, standard deviation = 3.8
Mean = 83, standard deviation = 9
Mean = 415, standard deviation = 6
Mean = 415, standard deviation = 9
Mean = 415, standard deviation = 19
27. A rock concert producer has scheduled an outdoor concert. If it is warm that day, she
expects to make a $20,000 profit. If it is cool that day, she expects to make a $5,000
profit. If it is very cold that day, she expects to suffer a $12,000 loss. Based upon
historical records, the weather office has estimated the chances of a warm day to be 0.60;
the chances of a cool day to be 0.25. What is the producer's expected profit?
(a)
(b)
(c)
(d)
(e)
$5,000
$13,000
$15,050
$13,250
$11,450
28. The scores on the Harrison High School AP Stat Test #1 (T1) had a mean of 27 with a
standard deviation of 3 and the scores on Test #2 (T2) had a mean of 29 with a standard
deviation of 4. To reflect the true brilliance of the students taking the course, the total
score had to be adjusted according to the following definition: Total = 2*T1+3*T2 . What is
the mean and standard deviation of Total if the scores on each test are independent?
(a)   141,   13.4
(b)   141,   18
(c)   141,   18.4
(d)   141,   43.1
(e) cannot be determined
29. When a life insurance company sells a 1-year policy, the policyholder pays a premium,
which is the amount of money paid to own the policy for 1 year. The insurance company
promises to pay a benefit to the policyholder’s survivors should the holder die during that
year, but the company keeps the premium. If the policyholder lives through the entire year,
no benefit is paid and the company keeps the premium.
If a policyholder pays a premium of $120 for a 1-year policy with a benefit of $50,000, and
the probability of the policyholder dying during the year is 0.0015, what is the expected
monetary net loss/gain for the insurance company?
(a)
(b)
(c)
(d)
(e)
A gain of about $150
A gain of about $45
Break even – no net gain or loss
A loss of about $45
A loss of about $150
30. In each of the millions of boxes of Jolly Cereal there is a toy. There are five different
kinds of toys of which Jimmy has four. Jimmy wants to know how many more boxes he
would expect to buy before he gets the fifth kind of toy. Why is this NOT a binomial
calculation?
(a)
(b)
(c)
(d)
(e)
There are five types of toys.
The number of boxes he will buy is not fixed.
The probability of getting the fifth type of toy is not 0.5.
The probability of getting the fifth toy is effectively constant.
The five types of toys are not known to be equally distributed.
31. The local scout troop is selling raffle tickets as a fundraiser. They have 500 tickets to
sell and the scouts charge $5.00 for each ticket. The distribution of the prizes, X, is
shown below:
X
Number of winning tickets
$0
461
$10
30
$50
6
$100
2
$200
1
If you purchase a ticket, what is the net expected gain/loss for your ticket?
(a)
(b)
(c)
(d)
(e)
Gain of $2.00
Gain of $3.00
Loss of $2.00
Loss of $3.00
Loss of $5.00
Part II – Free Response (Questions 1-2) – Please answer thoroughly.
1. Winston, Mr. League’s adorable pug, loves to play catch. Unfortunately, he (Winston, not
Mr. League) is not particularly adept at catching as his probability of catching the ball is
0.12.
I. Mr. League is interested in determining how many tosses it will take for Winston to
catch the ball once.
(i) Can this situation be described as binomial, geometric, or neither? Explain why.
(ii) What is the expected number of tosses it will take for Winston to catch the ball once?
(iii) What is the probability it will take exactly 10 tosses in order for Winston to catch the
ball?
II. Sergeant Slaughter, avid baseball player & coach, decides to train Winston. After
three-a-day training sessions for 4 weeks, the probability that Winston catches the
ball has increased to 0.42. Sergeant Slaughter is interested in determining the
number of times Winston will catch the ball in 25 tosses.
(i)
Can this situation be described as binomial, geometric, or neither? Explain why.
(ii)
What is the expected number of times that Winston will catch the ball?
(iii)
What is the probability that Winston will catch the ball 8 times in 25 tosses? Show
your work!
III. Mr. League, knowing that Winston is a reasonably smart dog, knows that his
probability of catching the ball will actually improve 0.01 after each toss. Mr. League
would like to find out the number of tosses required for Winston to catch the ball
three times.
(i) Can this situation be described as binomial, geometric, or neither? Explain why.
(ii) (BONUS!!!) If Winston’s initial probability of catching the ball is 0.12, what is the
probability that it will take five tosses for Winston to catch the ball three times?
2. For an upcoming concert, each customer may purchase up to 3 child tickets and 3 adult
tickets. Let C be the number of child tickets purchased by a single customer. The
probability distribution of the number of child tickets purchased by a single customer is
given in the table below.
C
P(C)
0
0.4
1
0.3
2
0.2
3
0.1
(a) Compute the mean and the standard deviation of C.
(b) Suppose the mean and the standard deviation of the number of adult tickets
purchased by a single customer is 2 and 1.2, respectively. Assume that the number
of child tickets and the number of adult tickets purchased are independent random
variables. Compute the mean and the standard deviation of the total number of
adult and child tickets purchased by a single customer.
(c) Suppose each child ticket costs $15 and each adult ticket costs $25. Compute the
mean and the standard deviation of the total amount spent per purchase.