Sec. 6.1 & 6.2 Review: Random Variables
AP Statistics
Name:
I. Multiple Choice.
1. A television game show has three payoffs with the following probabilities:
Payoff ($)
Probability
0
0.7
500
0.25
5,000
0.05
What are the mean and standard deviation of the payoff variable?
(a)
(b)
(c)
(d)
(e)
µ = 375, σ = 361
µ = 375, σ = 1,083
µ = 1,833, σ = 1,816
µ = 1,833, σ = 2,248
None of the above gives a set of correct answers.
2. An insurance company charges $900 annually for auto policies for teenagers. The policy specifies the company will
pay $1,500 for a minor accident and $8,000 for a major accident. If the probability of a teenager having a minor accident
during the year is .15 and of having a major accident is .05, how much can the insurance company expect to make on 10
policies?
(a) $275
(b) $625
(c) $2,750
(d) $6,250
(e) $9,000
3. Box A has four $10 bills and a single $100, box B has 400 $10 bills and 100 $100 bills, and box C has 28 $1 bills.
You can have all of box C or blindly pick one bill out of either box A or box B. Which choice offers the greatest expected
winning?
(a)
(b)
(c)
(d)
(e)
Box A
Box B
Box C
Either A or B, but not C
All offer the same expected winning.
4. Suppose the average outstanding loan for college graduates is $23,500 with a standard deviation of $7,200 and is
normally distributed. A college graduate is randomly selected, what is the probability that his mean outstanding loan is
under $21,000?
(a) 0.0000
(b) 0.6368
(c) 0.6528
(d) 0.3632
(e) 0.3472
5. Given a random variable X taking three possible values 𝑥1 , 𝑥2 , 𝑥3 , which of the following statements must be true?
I. 𝑥1 + 𝑥2 + 𝑥3 = 1
1
II. 𝐸(𝑋) = 3 ∑ 𝑥𝑖
III. 𝑣𝑎𝑟(𝑋) =
(a) I only
1
∑(𝑥𝑖
3
(b) III only
− 𝑥̅ )2
(c) II and III
(d) I, II, and III
(e) None of the statements are true.
6. The number of hybrid cars a dealer sells weekly has the following probability distribution:
Number of Hybrids
Probability
0
0.32
1
0.28
2
0.15
3
0.11
4
0.08
5
0.06
The dealer purchases the cars for $21,000 and sells them for $24,500. What is the expected weekly profit from selling
hybrid cars?
(a) $2,380
(b) $3,500
(c) $5,355
(d) $8,109
(e) $37,485
7. Three fair coins are tossed. If all land “heads,” the player wins $10, and if exactly two land heads, the player wins $5.
If it costs $4 to play, what is the player’s expected outcome after four games?
(a)
(b)
(c)
(d)
(e)
Loss of $0.875
Loss of $1.00
Loss of $3.50
Win of $2.25
Win of $9.00
Answers to Multiple Choice: 1. B, 2. C, 3. E, 4. D, 5. E, 6. C, 7. C
II. Free Response.
8. A spinner on a full circle can take on any decimal value between 0 and 400. What is the probability that the spinner
will land between 175 and 225? Hint: Draw a uniform distribution.
9. Now that the new models are here, a car dealership has lowered prices on last years’ models. An energetic salesperson
estimates the following probability distribution of X, the number of cars that she’ll sell next week.
X
P(X)
0
0.05
1
0.15
2
0.35
3
0.25
4
0.20
(a) Find and interpret the expected value and the standard deviation of X.
(b) Suppose that this salesperson earns a fixed weekly wage of $300 plus a $2000 commission for each car sold.
What are her expected weekly wages? (Let Y = weekly wages.) What is the standard deviation in her weekly
wages?
10. A local arcade is hosting a tournament in which contestants play an arcade game with possible scores ranging from 0
to 29. The arcade has set up multiple game tables so that all contestants can play the game at the same time; thus
contestant scores are independent. Each contestant’s score will be recorded as he or she finishes, and the contestant with
the highest score is the winner.
After practicing the game many times, Josephine, one of the contestants, has established the probability distribution of her
scores, shown in the table below.
Josephine’s Distribution
Score
16
Probability
0.10
17
0.30
18
0.40
19
0.20
Caroline, another contestant, has also practiced many times. The probability distribution for her scores is shown in the
table below.
Caroline’s Distribution
Score
17
Probability
0.45
18
0.40
19
0.15
(a) Calculate the expected score for each player.
(b) Suppose that Josephine scores 16 and Caroline scores 17. The difference (Josephine minus Caroline) of their
scores is -1. List all combinations of possible scores for Josephine and Caroline that will produce a difference
(Josephine minus Caroline) of -1, and calculate the probability for each combination.
(c) Find the probability that the difference (Josephine minus Caroline) in their scores is -1.
(d) The table below lists all the possible differences in the scores between Josephine and Caroline and some
associated probabilities.
Distribution (Josephine minus Caroline)
Difference
-3
-2
-1
Probability
0.015
0
0.325
1
0.260
2
0.090
Complete the table and calculate the probability that Caroline’s score will be higher than Josephine’s score.
11. The standard “Olympic Distance” triathlon is a 1.5 km swim, 40 km bike ride, and 10 km run. In a random sampling
of recent competitions, the mean and standard deviation of the participants’ times for each event were:
Mean (min)
Standard Deviation (min)
Swim
29
5
Bike Ride
90
10
Run
68
12
Assume the times for the three legs of the race are each normally distributed and independent. What is the probability that
a participant will complete the triathlon in less than 175 minutes?
12. You have two scales for measuring weights in a chemistry lab. Both scales give answers that vary a bit in repeated
weighings of the same item. If the true weight of a compound is 2.00 grams (g), the first scale produces readings X that
have a mean 2.000 g and standard deviation 0.002 g. The second scale’s readings Y have a mean 2.001 g and standard
deviation 0.001 g. Find the probability that if an item is weighed on both scales, its reading from the first the first scale
will be less than its reading from the second scale. Assume the scales are both Normally distributed and independent of
each other.