MDM4U Worksheet: Independent and Dependent Events
Multiple Choice
Identify the choice that best completes the statement or answers the
question.
1. What is the probability that three standard dice rolled simultaneously
will all land with the same the same number facing up?
1
1
1
1
a.
b.
c.
d.
6
36
216
3
2. Suppose you simultaneously roll a standard die and spin a spinner that
is divided into 10 equal sectors, numbered 1 to 10. What is the
probability of getting a 4 on both the die and the spinner?
1
1
1
1
a.
b.
c.
d.
4
16
15
60
3. Suppose you simultaneously roll a standard die and spin a spinner
with eight equal sectors, numbered 1 to 8. What is the probability of
both rolling an even number and spinning an odd number?
1
1
1
1
a.
b.
c.
d.
12
48
4
6
4. A coin is flipped and a card is drawn from a standard deck of cards.
What is the probability of getting both heads and a red face card?
3
1
3
3
a.
b.
c.
d.
26
112
52
13
5. Which of the following statements is false?
a. The product rule for independent events states that
𝑃(𝑋 𝑎𝑛𝑑 𝑌) = 𝑃(𝑥 ) ∙ 𝑃(𝑌).
b. If event X is dependent on event Y, then
𝑃(𝑋 | 𝑌) =
𝑃(𝑋 𝑎𝑛𝑑 𝑌)
𝑃(𝑌)
.
c. If event X is dependent on event Y, then
𝑃(𝑋 𝑎𝑛𝑑 𝑌) = 𝑃(𝑋) × 𝑃(𝑋 |𝑌).
d. If event X is dependent on event Y, then
𝑃(𝑋 𝑎𝑛𝑑 𝑌) = 𝑃(𝑌) × 𝑃(𝑋 |𝑌).
6. If Mike does his mathematics homework today, the probability that he
will do it tomorrow is 0.8. The probability that he will do his
mathematics homework today is 0.7. What are the odds that he will do it
both today and tomorrow?
a. 4:1 b. 8:7
c. 3:2
d. 14:11
7. A bag contains three green marbles and four black marbles. If you
randomly pick two marbles from the bag at the same time, what is the
probability that both marbles will be black?
4
2
3
5
a.
b.
c.
d.
7
7
7
7
8. What is the probability of rolling a total of 7 in two rolls of a standard
die if you get an even number on the first roll?
1
1
1
5
a.
b.
c.
d.
12
6
9
18
Short Answer
9. Lesley-Anne estimates that she has a 75% chance of passing physics
and an 80% chance of passing English. Assuming that {passing English}
and {passing Physics} are independent events,
a) What is the probability that Lesley-Anne will pass only one of these
two subjects?
b) What are the odds in favour of Lesley-Anne failing both subjects?
10. If a satellite launch has a 97% chance of success, what is the
probability of three consecutive successful launches?
11. Carrie is a kicker on her rugby team. She estimates that her chances
of scoring on a penalty kick during a game are 75% when there is no
wind, but only 60% on a windy day. If the weather forecast gives a 55%
probability of windy weather today, what is the probability of Carrie
scoring on a penalty kick in a match this afternoon?
12. A bag contains three white marbles, five green marbles, and two red
marbles. What are the odds in favour of randomly picking both red
marbles in the first two tries? Assume that the first marble picked is not
put back into the bag.
13. If the probability of the Rangers defeating the Eagles in a hockey
3
game is , what is the probability that the Rangers will win two
7
consecutive games against the Eagles?
14. Statesville has two computer-controlled traffic lights on the road
between the main street and the highway. The probability of getting a
red light at the first traffic light is 0.45, and the probability of getting a
red light at the second one is 0.20 if you had been stopped by a red light
at the first one. What is the probability of being stopped by red lights at
both intersections?
Problems
15. A survey at a school asked students if they were ill with a cold or the
flu during the last month. The results were as follows. None of the
students had both a cold and the flu.
Cold Flu Healthy
Females 32
18
47
Males
25
19
38
Use these results to estimate the probability that
a) a randomly selected student had a cold in the last month
b) A randomly selected female student was healthy last month
c) A randomly selected student who had the flu last month is male
d) A randomly selected male student had either a cold or the flu last
month
16. To get out of jail free in the board game MONOPOLY®, you have
to roll doubles with a pair of standard dice. Determine the odds in favour
of getting out of jail on your first or second roll.
17. At an athletic event, athletes are tested for steroids using two
different tests. The first test has a 93.0% probability of giving accurate
results, while the second test is accurate 87.0% of the time. For a sample
that does contain steroids, what is the probability that
a) Neither test shows that steroids are present?
b) Both tests show that steroids are present?
c) At least one of the tests detects the steroids?
18. A test for the presence of E. coli in water detects the bacteria 97% of
the time when the bacteria is present, but also gives a false positive 2%
of the time, wrongly indicating the presence of E. coli in uninfected
water. If 10% of the water samples tested contain E. coli, what is the
probability that a test result indicating the presence of the bacteria is
accurate?
19. A study on the effects that listening to loud music through
headphones had on teenagers’ hearing found that 12% of those teenagers
in the sample who did listen to music in this way showed signs of
hearing problems. If 60% of the sample reported that they listened to
loud music on headphones regularly, and 85% of the sample were found
not to have hearing problems, are the events {having hearing problems}
and {listening to loud music on headphones} independent? Explain your
reasoning.
Answer Section
MULTIPLE CHOICE
1. B 2. D 3. C 4. C
5. C 6. D 7. B 8. B
SHORT ANSWER
9. a) 0.35
b) 1:19
10. About 91%
11.
12. The probability of picking the two red marbles in the first two picks
is
2
1
1
× = .
10
9
45
Therefore, the odds in favour of picking the two red marbles are 1:44.
13.
14. 9.0%
PROBLEM
15.
a)
b) Using the conditional probability formula, we get:
c) Restricting the sample space to only those who had the flu,
d) Restricting the sample space to only males,
16. The probability of rolling doubles on the first roll is
5
6
. The
36
probability of not rolling doubles on the first roll is . Therefore, the
6
probability of rolling doubles on the second roll is
The probability of rolling doubles on the first roll or the second roll is
1
5
11
+
=
6
36
36
Thus, the odds in favour of getting out of jail on either the first or second
try are 11:25.
17.
18. If 10% of the water samples contain E. coli and the test is 97%
effective, then
However, 90% of the samples do not contain E. coli and these samples
will test positive 2% of the time, so
The overall probability of a positive test result is 0.097 + 0.018 = 0.115.
Therefore, the conditional probability formula gives
The probability is 84% that a positive test result is accurate.
19. If events A and B are independent, then
Since P(hearing problems) = 0.15 and P(listening to loud music on
headphones) = 0.60, then
However, the observed probability of having hearing problems and
listening to loud music on headphones is 0.12, which is significantly
higher than 0.09. Therefore, these two events cannot be independent if
the survey results are accurate.