Chapter 5—Probability
MULTIPLE CHOICE
1. An approach of assigning probabilities that assumes that all outcomes of the experiment are equally
likely is referred to as the:
A.
subjective approach.
B.
objective approach.
C.
classical approach.
D.
relative frequency approach.
2. If P(A) = 0.74, P(B) =0.76 and P(A B) =0.80, then P(A B) is:
A.
0.70.
B.
0.74.
C.
0.80.
D.
0.02.
3. If P(A) = 0.35, P(B) = 0.45 and P(A
B) =0.25, then P(A | B) is:
A.
1.4.
B.
1.8.
C.
0.714.
D.
0.556.
4. If P(A) = 0.20, P(B) = 0.30 and P(A
B) = 0.06, then A and B are:
A.
dependent events.
B.
independent events.
C.
mutually exclusive events.
D.
complementary events.
5. If A and B are mutually exclusive events with P(A) = 0.70, then P(B):
A.
can take any value between 0 and 1.
B.
can take any value between 0 and 0.70.
C.
cannot be larger than 0.30.
D.
cannot be determined from the information given.
6. If A and B are independent events with P(A) = 0.60 and P(A/B) = 0.60, then P(B) is:
A.
1.20.
B.
0.60.
C.
0.36.
D.
P(B) cannot be determined with the information given.
7. If P(A) = 0.60, P(B) = 0.58, and P(A B) = 0.76, then P(A B) is:
A.
0.60.
B.
0.76.
C.
0.16.
D.
0.44.
8. If you roll a fair (unbiased) die 60 times, you should expect an odd number to appear:
A.
in each of the first 30 rolls.
B.
30 out of the 60 rolls.
C.
on every other roll.
D.
40 out of the 60 rolls.
9. The collection of all possible outcomes of an experiment is called:
A.
a simple event.
B.
a sample space.
C.
a sample.
D.
a population.
10. Which of the following is not an approach to assigning probabilities?
A.
Classical approach
B.
Trial-and-error approach
C.
Relative frequency approach
D.
Subjective approach
11. A useful graphical method of constructing the sample space for an experiment is:
A.
a tree diagram.
B.
a pie chart.
C.
a histogram.
D.
an ogive.
12. When a fair die is rolled once, the sample space consists of the following six outcomes: 1, 2, 3, 4, 5, 6.
Given this sample space, which of the following is a simple event?
A.
Even number.
B.
6.
C.
Less than 1.
D.
More than 4.
13. Suppose P(A) = 0.35. The probability of complement of A is:
A.
0.35.
B.
0.50.
C.
0.65.
D.
–0.35.
14. An experiment consists of tossing three unbiased coins simultaneously. Drawing a probability tree for
this experiment will show that the number of simple events in this experiment is:
A.
3.
B.
6.
C.
9.
D.
None of the above answers is correct.
15. If the events A and B are independent, with P(A) = 0.30 and P(B) = 0.40, then the probability that both
events will occur simultaneously is:
A.
0.10.
B.
0.12.
C.
0.70.
D.
0.75.
16. Two events A and B are said to be independent if:
A.
P(A B) = P(A) P(B).
B.
P(B | A) = P(B).
C.
P(A | B) = P(A).
D.
All of the above answers are correct.
17. Two events A and B are said to mutually exclusive if:
A.
P(A | B) = 1.
B.
P(B | A) =1.
C.
P(A
B) =1.
D.
P(A
B) = 0.
18. Which of the following statements is always correct?
A.
P(A
B) = P(A) P(B).
B.
P(A
B) = P(A) + P(B).
C.
P(A
B) = P(A) + P(B) + P(A
B).
D.
P(A) = 1 – P( ).
19. Which of the following is a requirement of the probabilities assigned to the outcomes
i
O
?
A.
P(Oi) 0.
B.
P(Oi) 0.
C.
0 P(Oi) 0, for each i.
D.
P(Oi) = 1 + P(i).
20. An experiment consists of three stages. There are two possible outcomes in the first stage, three
possible outcomes in the second stage, and four possible outcomes in the third stage. Drawing a tree
diagram for this experiment will show that the total number of outcomes is:
A.
9.
B.
24.
C.
26.
D.
18.
21. If A and B are mutually exclusive events, with P(A) = 0.20 and P(B) = 0.30, then P(A
B) is:
A.
0.50.
B.
0.10.
C.
0.00.
D.
0.06.
22. If A and B are independent events, with P(A) = 0.60 and P(B) = 0.70, then the probability that A occurs
or B occurs or both occur is:
A.
1.30.
B.
0.88.
C.
0.42.
D.
0.10.
23. If A and B are mutually exclusive events, with P(A) = 0.30 and P(B) = 0.40, then P(A
B) is:
A.
0.10.
B.
0.12.
C.
0.70.
D.
None of the above answers is correct.
24. If A and B are independent events, with P(A) = 0.20 and P(B) =0.60, then P(A | B) is:
A.
0.20.
B.
0.60.
C.
0.40.
D.
0.80.
25. If P(A) = 0.25 and P(B) = 0.65, then P(A
B) is:
A.
0.25.
B.
0.40.
C.
0.90.
D.
P(A
B) cannot be determined from the information given.
26. If a coin is tossed three times, and a statistician predicts that the probability of obtaining three heads in
a row is 0.125, which of the following assumptions is irrelevant to his prediction?
A.
The events are dependent.
B.
The events are independent.
C.
The coin is unbiased.
D.
All of the above assumptions are relevant to his prediction.
27. If an experiment consists of five outcomes, with
1
( )P O =
0.10,
2
( )P O =
0.10,
3
( )P O =
0.30,
4
( )P O =
0.25, then
5
( )P O
is:
A.
0.75.
B.
0.25.
C.
0.20.
D.
0.80.
28. Of the last 400 customers entering a supermarket, 20 have purchased a mobile phone. If the classical
approach for assigning probabilities is used, the probability that the next customer will purchase a
mobile phone is:
A.
0.80.
B.
0.20.
C.
0.05.
D.
0.50.
TRUE/FALSE
1. If the event of interest is A, the probability that A will not occur is the complement of A.
2. The probability of event A and event B occurring must be equal to 1.
3. The relative frequency approach to probability depends on the law of large numbers.
4. The annual estimate of the number of deaths of infants is an example of the classical approach to
probability.
5. The classical approach to assigning probability can be applied for experiments that have equally likely
outcomes.
6. Based on past exam results in principles of accounting you estimate that there is an 83% chance of
passing the exam. This is an example of the subjective approach to probability.
7. Probability refers to a number between 0 and 1 (inclusive), which expresses the chance that an event
will occur.
8. If event A does not occur, then its complement must occur.
9. Marginal probability is the probability that a given event will occur, with no other events taken into
consideration.
10. Conditional probability is the probability that an event will occur, given that another event will also
occur.
11. If we wished to determine the probability that one or more of several events will occur in an
experiment, we would use addition rules.
12. Two or more events are said to be independent when the occurrence of one event has no effect on the
probability that another will occur.
13. Five students from a statistics class have formed a study group. Each may or may not attend a study
session. Assuming that the members will be making independent decisions on whether or not to attend,
there are 32 different possibilities for the composition of the study session.
14. When events are mutually exclusive, they cannot happen at the same time.
15. According to an old song lyric, ‘love and marriage go together like a horse and carriage’. Let love be
event A and marriage be event B. Events A and B are mutually exclusive.
16. When it is not reasonable to use the classical approach to assigning probabilities to the outcomes of an
experiment, and there is no history of the outcomes, we have no alternative but to employ the
subjective approach.
17. Bayes’ theorem allows us to compute conditional probabilities from other forms of probability.
18. A useful graphical method of constructing the sample space for an experiment is the pie chart.
19. An experiment consists of tossing three fair (unbiased) coins simultaneously. This experiment has
eight possible outcomes.
20. Assume that A and B are independent events, with P(A) = 0.30 and P(B) = 0.50. The probability that
both events will occur simultaneously is 0.80.
21. Two events A and B are said to be independent if P(A
B) = P(A) P(B).
22. Two events A and B are said to mutually exclusive if P(A) = P(B).
23. If events A and B have nonzero probabilities, then they can be both independent and mutually
exclusive.
24. If A and B are independent events, with P(A) = 0.30 and P(B) = 0.50, then P(B | A) is 0.50.
25. An effective and simple method of applying the probability rules is the probability tree, wherein the
events of an experiment are represented by lines.
26. The probability of the union of two mutually exclusive events A and B is P(A
B) = P(A) + P(B).
27. The relative frequency approach is useful in interpreting probability statements such as those heard
from weather forecasters or scientists.
28. Given that events A and B are independent, and that P(A) = 0.9 and P(B | A) = 0.5, then P(A
B) =
0.45.
29. Jim and John go to a coffee shop during their lunch break and toss a coin to see who will pay. The
probability that John will pay three days in a row is 0.125.
SHORT ANSWER
1. Three candidates for the presidency of a university’s student union, Alice, Brenda and Cameron, are to
address a student forum. The forum’s organiser is to select the order in which the candidates will give
their speeches, and must do so in such a way that each possible order is equally likely to be selected.
a. What is the random experiment?
b. List the simple events in the sample space.
c. Assign probabilities to the simple events.
d. What is the probability that Cameron will speak first?
e. What is the probability that one of the women will speak first?
f. What is the probability that Alice will speak before Cameron does?
2. Suppose A and B are two independent events, with P(A) = 0.30 and P(B) = 0.60.
a. Find P(A | B).
b. Find P(B | A).
c. Find P(A and B).
d. Find P(A or B).
3. A PhD graduate has applied for a job with two universities, A and B. The graduate feels that she has a
60% chance of receiving an offer from university A, and a 50% chance of receiving an offer from
university B. If she receives an offer from university B, she believes that she has an 80% chance of
receiving an offer from university A.
a. What is the probability that both universities will make her an offer?
b. What is the probability that at least one university will make her an offer?
c. If she receives an offer from university B, what is the probability that she will not receive an offer
from university A?
4. There are three approaches to determining the probability that an outcome will occur: the classical,
relative frequency, and subjective approaches. Which is most appropriate in determining the
probability of the following outcomes?
a. In the next lotto draw the winning numbers will be 1, 2, 3, 4 and 5.
b. 18 students out of 21 in a business statistics class will pass the final exam.
c. There will be no major war in Africa next year.
d. Two of the next 10 new cars sold in Melbourne will be European made.
5. Suppose P(A) = 0.10, P(B) = 0.70, and P(B/A) = 0.30.
a. Find P(A
B).
b. Find P(A
B).
c. Find P(A | B).
6. At the beginning of each year, an investment newsletter predicts whether or not the stock market will
rise over the coming year. Historical evidence reveals that there is a 75% chance that the stock market
will rise in any given year. The newsletter has predicted a rise for 80% of the years when the market
actually rose, and has predicted a rise for 40% of the years when the market fell. Find the probability
that the newsletter’s prediction for next year will be correct.
7. Suppose P() = 0.20, P( | A) = 0.40, and P( | ) = 0.50.
a. Find P(A).
b. Find P( A).
c. Find P( ).
d. Find P().
8. A standard admissions test was given at three locations. One thousand students took the test at location
A, 600 students at location B, and 400 students at location C. The percentages of students from
locations A, B and C who passed the test were 70%, 68% and 77%, respectively. One student is
selected at random from among those who took the test.
a. What is the probability that the selected student passed the test?
b. If the selected student passed the test, what is the probability that the student took the test at
location B?
c. What is the probability that the selected student took the test at location C and failed?
9. The following table shows the numbers of cars sold by a car dealer during the last 30 weeks.
Number of cars sold
Number of weeks
2
5
3
9
4
10
5
4
6
2
a. Define the random variable of interest to the dealer.
b. List the simple events in the sample space.
c. Assign probabilities to the simple events and show the probability distribution.
d. What approach have you used in determining the probabilities in part (c)?
e. What is the probability of selling no more than four cars in any given week?
10. A woman is expecting her second child. Her doctor has told her that she has a 50–50 chance of having
another girl. If she has another girl, there is a 90% chance that she will be taller than the first. If she
has a boy, however, there is only a 25% chance that he will be taller than the first child. Find the
probability that the woman’s second child will be taller than the first.
11. A survey of a magazine’s subscribers indicates that 50% own a home, 80% own a car, and 90% of the
homeowners who subscribe also own a car. What proportion of subscribers:
a. own both a car and a house?
b. own a car or a house, or both?
c. own neither a car nor a house?
12. Suppose A and B are two mutually exclusive events for which P(A) = 0.35 and P(B) = 0.60.
a. Find P(A
B).
b. Find P(A
B).
c. Find P(B | A).
d. Are A and B independent events? Explain using probabilities.
13. Suppose P(A) = 0.10, P(B | A) = 0.20, and P(B | ) = 0.40.
a. Find P(A
B).
b. Find P(A
B).
c. Find P().
d. Find P(A or B).
14. An investor tells you that in his estimation there is a 75% chance that a particular stock’s price will rise
over the next three weeks.
a. Which approach was used to produce this figure?
b. Interpret the 75% probability.
15. The sample space of the toss of a fair die is S = {1, 2, 3, 4, 5, 6}. If the die is balanced, each simple
event has the same probability. Find the probability of the following events.
a. Equal to 4.
b. A number greater than or equal to 2.
c. A number greater than 5.
d. A number between 2 and 4, inclusive.
16. Suppose P(A) = 0.4, P(B) = 0.5, and P(A
B) = 0.
a. Find P(A
B).
b. Are A and B mutually exclusive events? Explain.
c. Are A and B independent events? Explain.
17. Suppose P(A) = 0.30, P(B) = 0.50, and P(B /A) = 0.60.
a. Find P(A
B).
b. Find P(A
B).
c. Find P(A /B).
18. Is it possible to have two events for which P(A) = 0.40, P(B) = 0.50, and P(A
B) = 0.30? Explain.
19. A statistics professor classifies his students according to their gender and the number of hours of paid
work they do a week. The following table gives the proportions of students falling into the various
categories. One student is selected at random.
Paid Work (hours/week)
Gender
0
1–8
9–16
Over 16
Male
0.06
0.20
0.15
0.06
Female
0.12
0.18
0.20
0.03
a. If the student selected is male, what is the probability that he works between 1 and 8 hours a
week?
b. If the selected student works more than 16 hours a week, what is the probability that the student is
female?
c. What is the probability that the student selected is female or does do any paid work or both?
d. Is gender independent of the number of hours of paid work done a week? Explain using
probabilities.
20. A pharmaceutical firm has discovered a new diagnostic test for a certain disease that has infected 1%
of the population. The firm has announced that 95% of those infected will show a positive test result,
while 98% of those not infected will show a negative test result. What proportion of test results are
correct?
ANS: