AP Statistics
Extra Practice
Name: ____________
Date: _____________
7.1A: Discrete Random Variables
1.
Let x be the number of courses for which a randomly selected student at a certain university is
registered. The probability distribution of x appears in the table below:
x
P(x)
1
.02
2
.03
3
.09
4
.25
5
.40
6
.16
7
.05
What is 𝑃(𝑥 = 4) ?
What is 𝑃(𝑥 ≤ 4) ?
What is the probability that the selected student is taking at most five courses?
What is the probability that the selected student is taking at least five courses?
More than five courses?
e. Calculate 𝑃(3 ≤ 𝑥 ≤ 6) and 𝑃(3 < 𝑥 < 6).
Explain in words why these two probabilities are different.
a.
b.
c.
d.
2.
A restaurant has four bottles of a certain wine in stock. Unbeknownst to the wine steward, two
of these bottles (numbers 1 and 2) are bad. Suppose that two bottles are ordered, and let x be
the number of good bottles among these two.
a. One possible experimental outcome is (1,2) [bottles 1 and 2 are the ones selected] and
another is (2,4). List all possible outcomes.
b. Assuming that the two bottles are randomly selected from among the four (equally likely
outcomes), what is the probability of each outcome in part a.?
c. The value of x for the (1,2) outcome is 0 (neither selected bottle is good), and x = 1 for the
outcome (2,4). Determine the x value for each possible outcome. Then use the
probabilities in part b. to determine the probability distribution of x.
3.
Let y denote the number of broken eggs in a randomly selected carton of one dozen “store
brand” eggs at a certain market. Suppose that the probability distribution of y is as follows:
y
P(y)
0
.65
1
.20
2
.10
3
.04
4
?
a. Only y values 0, 1, 2, 3, and 4 have positive probabilities. What is P(y =4) ?
b. How would you interpret P(y = 1) ?
c. Calculate 𝑃(𝑦 ≤ 2), the probability that the carton contains at most two broken eggs, and
interpret this probability.
d. Calculate 𝑃(𝑦 < 2), the probability that the carton contains fewer than two broken eggs.
Why is this smaller than the probability in part c ?
e. What is the probability that the carton contains exactly ten unbroken eggs?
[Hint: What is the corresponding value of y ?]
f. What is the probability that at least ten eggs are unbroken?
4.
Airlines sometimes overbook flights. Suppose that for a plane with 100 seats, an airline takes
110 reservations. Define the variable x as the number of people who actually show up for a
sold-out flight. From past experience, the probability distribution of x is given in the following
table:
x
P(x)
95
.05
96
.10
97
.12
98
.14
99
.24
100
.17
101
.06
x
P(x)
103
.03
104
.02
105
.01
106
.005
107
.005
108
.005
109
.0037
102
.04
110
.0013
a. What is the probability that the airline can accommodate everyone who shows up for the
flight?
b. What is the probability that not all passengers can be accommodated?
c. If you are trying to get a seat on such a flight and you are number 1 on the standby list,
what is the probability that you will be able to take the flight? What if you are number 3?
5.
Many manufacturers have quality control programs that include inspection of incoming
materials for defects. Suppose that a computer manufacturer receives computer boards in lots
of five. Two boards are selected from each lot for inspection. We can represent possible
outcomes of the selection process by pairs. For example, the pair (1,2) represents the
selection of boards 1 and 2 for inspection.
a. List the ten different possible outcomes.
b. Suppose that boards 1 and 2 are the only defective boards in a lot of five. Two boards are
to be chosen at random. Define x to be the number of defective boards observed among
those inspected. Find the probability distribution of x.
6.
A box contains five slips of paper, marked $1, $1, $1, $10 and $25. The winner of a contest
will select two slips of paper at random and will then get the larger of the dollar amounts on the
two slips. Define a random variable w by
w = amount awarded
Determine the probability distribution of w. [Hint: Think of the slips as numbered 1, 2, 3, 4,
and 5, so that an outcome of the experiment consists of two of these numbers.]
7.
Components coming off an assembly line are either free of defects (S, for success) or
defective (F, for failure). Suppose that 70% of all such components are defect-free.
Components are independently selected and tested one by one. Let y denote the number of
components that must be tested until a defect-free component is obtained.
a. What is the smallest possible y-value, and what experimental outcome gives this y-value?
What is the second smallest y-value, and what outcome gives rise to it?
b. What is the set of all possible-y-values?
c. Determine the probability of each of the five smallest y-values. You should see a pattern
that leads to a simple formula for P(y), the probability distribution of y.