Name: ________________________ Class: ___________________ Date: __________
ID: A
Probability Sample Test
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
____
____
____
____
____
____
____
____
1. A coin is tossed three times. What is the probability of tossing exactly two heads?
5
1
1
3
b.
c.
d.
a.
8
2
4
8
2. A standard die is rolled. What is the probability of rolling a 4?
1
1
1
1
a.
b.
c.
d.
4
3
2
6
3. A computerized random number generator generates a number between 1 and 100 for a lottery. What is the
probability that it is divisible by 3?
3
1
33
1
a.
b.
c.
d.
100
100
100
3
4. Which of the following statements is false?
a. “Theoretical probability” is also called “classical probability” or “a priori probability.”
b. “Experimental probability” is another term for “relative-frequency probability.”
c. A “sample space” is the set of all possible outcomes of a probability experiment.
d. “Empirical probability” is another term for “subjective probability.”
5. Two identical spinners each have five equal sectors that are numbered 1 to 5. What is the probability of a
total less than 9 when you spin both these spinners?
3
6
22
4
a.
b.
c.
d.
25
25
25
5
6. If the odds in favour of snow tomorrow are 4:7, what is the probability of snow tomorrow?
4
7
4
7
b.
c.
d.
a.
7
11
11
4
7. Two dice are rolled and the upper faces are recorded. What are the odds against the sum of the two dice
being 3?
a. 18:1
b. 1:18
c. 17:1
d. 1:17
8. Twelve books are placed on a shelf in random order. What is the probability that their titles are in
alphabetical order?
1
1
1
1
a.
b.
c.
d.
12
12!
26 C 12
26 P 12
9. Five different letters are randomly selected from the alphabet. What is the probability that they are A, B, C, D
and E?
1
1
5
5!
a.
b.
c.
d.
26 C 5
26 P 5
26 C 5
26 C 5
____ 10. Eight children line up for a photograph. What is the probability that they will be in ascending order of their
ages?
1
1
1
8
a.
b.
c.
d.
8!
8
8!
8 C8
1
Name: ________________________
ID: A
____ 11. Five cards are laid face up on a table. What is the probability that the first card is a king and the last card is
an ace?
8
4 P1 × 4 P1
4×4
4 P1 × 4 P1
b.
c.
d.
a.
52 P 2
52 C 2
8 P2
52 P 5
____ 12. Six people were asked to randomly select a food from a list of 20 foods. What is the probability that at least
two people select the same food?
a.
1−
P(20,6)
6
b.
P(20,6)
6
c.
1−
20
P(20,6)
d.
20
P(20,6)
20
20
____ 13. Which of the following pairs of events are dependent?
a. rolling a die; dealing a card from a standard deck
b. rolling a 6 on one die; rolling a 6 on a second die
c. rolling doubles with two dice; rolling an even sum with two dice
d. dealing a king from a standard deck; spinning a 3 on a spinner
____ 14. Which of the following pairs of events are independent?
a. being dealt the king of hearts from a standard deck; being dealt the king of clubs in the
same hand
b. selecting the winning lottery number; selecting the second place lottery number on a
different ticket
c. the first coin shows heads; three out of four coins show heads
d. the first person selected is male; the fifth person selected is female
____ 15. Of all the students at Eastern Collegiate, 55% were born in Canada. Of those born in Canada, 85% speak
English at home. Of those not born in Canada, 32% speak English at home. A student was selected at
random. What is the probability that this student was born in Canada or speaks English at home?
c. 0.55 + 0.32
a. 0.55 × 0.85 × 0.32
b. 0.55 + 0.45 × 0.32
d. 0.55 × 0.85 + 0.32
____ 16. A newspaper sports reporter has a 58% accuracy for predicting the winners in NHL hockey games. A radio
sports reporter has a 65% accuracy for predicting the winners. For a particular game, what is the probability
that at least one of these reporters will make a correct prediction?
a. 1 – 0.58 × 0.65
c. 0.58 + 0.65 – 0.58 × 0.65
b. 0.58 + 0.65
d. 0.58 × 0.65
____ 17. When two events are mutually exclusive, it means that
a. there is an intersection between the two events
b. the probability of one event depends on the probability of the other
c. the two events cannot occur at the same time
d. the two events are equally likely
____ 18. If P(A) = 0.4 and P(B) = 0.2, and A and B are mutually exclusive events, which one of the following
statements is true?
a. P(A and B) = 0.6
c. P(A and B) = 0
b. P(A or B) = 0.08
d. P(A or B) = 0.2
2
Name: ________________________
ID: A
Consider the Venn diagram below, with the number of items in each set indicated. One item is selected from
the universal set.
____ 19. Which is a true statement?
a. Events A and B are mutually exclusive.
b. Events A and B are non-mutually exclusive.
c. Events A and B are dependent.
d. Event B is the complement of event A.
____ 20. P(A or B) =
9
6
a.
b.
c.
31
31
25
31
d.
0
Short Answer
21. Explain the meaning of the terms in the probability formula, P(A) =
n(A)
.
n(S)
22. A coin is tossed three times. What is the probability of tossing at least one head?
23. A standard die is rolled. What is the probability of rolling a prime number?
24. Two standard dice are rolled. What is the probability that a sum less than 7 is not rolled?
25. What are the odds in favour of July 1st being a Tuesday?
26. If the odds are 9 : 1 against the next car you see being red, what percent of cars in your area are red?
27. A club with eight members from grade 11 and five members from grade 12 is to elect a president,
vice-president, and secretary. What is the probability that grade 12 students will be elected for all three
positions, assuming that all club members have an equal chance of being elected? Give your answer as a
percent, rounded to one decimal place.
28. A four-member curling team is randomly chosen from six grade-11 students and nine grade-12 students.
What is the probability that the team has at least one grade-11 student?
29. Four students are selected at random from a group of three girls and five boys. What is the probability that
two girls and two boys will be selected?
3
Name: ________________________
ID: A
30. Lesley-Anne estimates that she has a 75% chance of passing physics and an 80% chance of passing English.
Assume 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?
31. If a satellite launch has a 97% chance of success, what is the probability of three consecutive successful
launches?
32. The probability that Jacqueline will be elected to the students’ council is 0.6, and the probability that she will
be selected to represent her school in a public-speaking contest is 0.75. The probability of Jacqueline
achieving both of these goals is 0.5.
a) Are these two goals mutually exclusive? Explain your answer.
b) What is the probability that Jacqueline is either elected to the students’ council or picked for the
public-speaking contest?
c) What is the probability that she fails to be selected for either the students’ council or the public-speaking
contest?
33. The probability that Sarjay will play golf today is 60%, the probability that he will play golf tomorrow is
75%, and the probability that he will play golf on both days is 50%. What is the probability that he does not
play golf on either day?
Problem
34. Suppose you randomly draw two marbles, without replacement, from a bag containing six green, four red,
and three black marbles.
a) Draw a tree diagram to illustrate all possible outcomes of this draw.
b) Determine the probability that both marbles are red.
c) Determine the probability that you pick at least one green marble.
35. 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.
Females
Males
Cold
32
25
Flu
18
19
Healthy
47
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 is male given that they have the flu
d) a randomly selected student has a cold or flue given that the student is male
36. 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.
4
ID: A
Probability Sample Test
Answer Section
MULTIPLE CHOICE
1. ANS:
OBJ:
KEY:
2. ANS:
OBJ:
KEY:
3. ANS:
OBJ:
KEY:
4. ANS:
OBJ:
KEY:
5. ANS:
OBJ:
KEY:
6. ANS:
OBJ:
KEY:
7. ANS:
OBJ:
KEY:
8. ANS:
OBJ:
KEY:
9. ANS:
OBJ:
KEY:
10. ANS:
OBJ:
KEY:
11. ANS:
OBJ:
KEY:
12. ANS:
OBJ:
KEY:
13. ANS:
OBJ:
KEY:
14. ANS:
OBJ:
KEY:
D
PTS: 1
DIF:
Section 6.1 LOC: A1.3
TOP:
probability
D
PTS: 1
DIF:
Section 6.1 LOC: A1.3
TOP:
probability
C
PTS: 1
DIF:
Section 6.1 LOC: A1.3
TOP:
probability
D
PTS: 1
DIF:
Section 6.1 LOC: A1.1
TOP:
probability
C
PTS: 1
DIF:
Section 6.1 LOC: A1.3
TOP:
probability
C
PTS: 1
DIF:
Section 6.2 LOC: A1.5
TOP:
odds and probability
C
PTS: 1
DIF:
Section 6.2 LOC: A1.5
TOP:
odds
B
PTS: 1
DIF:
Section 6.3 LOC: A2.5
TOP:
probability using counting techniques
B
PTS: 1
DIF:
Section 6.3 LOC: A2.5
TOP:
probability using counting techniques
A
PTS: 1
DIF:
Section 6.3 LOC: A2.5
TOP:
probability using counting techniques
D
PTS: 1
DIF:
Section 6.3 LOC: A2.5
TOP:
probability using counting techniques
A
PTS: 1
DIF:
Section 6.3 LOC: A2.5
TOP:
probability using counting techniques
C
PTS: 1
DIF:
Section 6.4 LOC: A1.6
TOP:
dependent and independent events
B
PTS: 1
DIF:
Section 6.4 LOC: A1.6
TOP:
dependent and independent events
1
1
REF: Knowledge & Understanding
Counting and Probability
1
REF: Knowledge & Understanding
Counting and Probability
1
REF: Knowledge & Understanding
Counting and Probability
1
REF: Knowledge & Understanding
Counting and Probability
1
REF: Knowledge & Understanding
Counting and Probability
1
REF: Knowledge & Understanding
Counting and Probability
1
REF: Knowledge & Understanding
Counting and Probability
2
REF: Application
Counting and Probability
2
REF: Application
Counting and Probability
2
REF: Application
Counting and Probability
3
REF: Application
Counting and Probability
2
REF: Application
Counting and Probability
2
REF: Knowledge & Understanding
Counting and Probability
2
REF: Knowledge & Understanding
Counting and Probability
ID: A
15. ANS:
OBJ:
KEY:
16. ANS:
OBJ:
KEY:
17. ANS:
OBJ:
KEY:
18. ANS:
OBJ:
KEY:
19. ANS:
OBJ:
KEY:
20. ANS:
OBJ:
KEY:
B
PTS: 1
Section 6.5 LOC: A1.6
dependent and independent events
C
PTS: 1
Section 6.4 LOC: A1.6
dependent and independent events
C
PTS: 1
Section 6.5 LOC: A1.5
mutually exclusive events
C
PTS: 1
Section 6.5 LOC: A1.5
mutually exclusive events
A
PTS: 1
Section 6.5 LOC: A1.5
mutually exclusive events
C
PTS: 1
Section 6.5 LOC: A1.5
mutually exclusive events
DIF: 2
REF: Application
TOP: Counting and Probability
DIF: 2
REF: Application
TOP: Counting and Probability
DIF: 1
REF: Knowledge & Understanding
TOP: Counting and Probability
DIF: 1
REF: Knowledge & Understanding
TOP: Counting and Probability
DIF: 1
REF: Knowledge & Understanding
TOP: Counting and Probability
DIF: 1
REF: Knowledge & Understanding
TOP: Counting and Probability
SHORT ANSWER
21. ANS:
P represents probability.
A represents the event A.
The letter n stands for “number” and n(A) represents the number of outcomes in which event A occurs.
S represents the sample space, and n(S) is the number of outcomes in the sample space (the total number of
possible outcomes).
PTS: 1
DIF: 1
OBJ: Section 6.1 LOC: A1.1
KEY: probability concepts
22. ANS:
7
8
REF: Communication
TOP: Counting and Probability
PTS: 1
OBJ: Section 6.1
KEY: probability
23. ANS:
1
2
REF: Knowledge & Understanding
TOP: Counting and Probability
PTS: 1
LOC: A1.3
DIF: 2
LOC: A1.3 | A1.5
DIF: 2
REF: Application
TOP: Counting and Probability
2
OBJ: Section 6.1
KEY: probability
ID: A
24. ANS:
7
12
PTS: 1
LOC: A1.5
25. ANS:
1:6
DIF: 2
REF: Application
TOP: Counting and Probability
OBJ: Section 6.1
KEY: complement
PTS: 1
LOC: A1.5
26. ANS:
10%
DIF: 1
REF: Application
TOP: Counting and Probability
OBJ: Section 6.2
KEY: odds
PTS: 1
LOC: A1.5
27. ANS:
60
5 P3
=
1716
13 P 3
DIF: 2
REF: Application
TOP: Counting and Probability
OBJ: Section 6.2
KEY: odds
PTS: 1
DIF: 2
REF: Application
LOC: A2.5
TOP: Counting and Probability
KEY: probability using counting techniques
28. ANS:
9 C4
126
1−
= 1−
1365
15 C 4
OBJ: Section 6.3
≈ 3.5%
=
1239
1365
PTS: 1
DIF: 3
REF: Application
LOC: A2.5
TOP: Counting and Probability
KEY: probability using counting techniques
29. ANS:
3 C2 × 5 C2
30
=
70
8 C4
=
OBJ: Section 6.3
3
7
PTS: 1
DIF: 2
REF: Application
LOC: A2.5
TOP: Counting and Probability
KEY: probability using counting techniques
3
OBJ: Section 6.3
ID: A
30. ANS:
a) 0.75 × (1 – 0.8) + (1 – 0.75) × 0.8 = 0.35
b) P(failing both) = 0.25 × 0.2
= 0.05
Odds in favour = 5 : 95
= 1 : 19
PTS: 1
DIF: 3
REF: Application
LOC: A1.6
TOP: Counting and Probability
31. ANS:
3
(0.97) = 0.912 673
OBJ: Section 6.4
KEY: dependent and independent events
PTS: 1
DIF: 2
REF: Application OBJ: Section 6.4
LOC: A1.6
TOP: Counting and Probability
KEY: dependent and independent events
32. ANS:
a) The two goals cannot be mutually exclusive since the probability of achieving both is 0.5.
b) 0.6 + 0.75 – 0.5 = 0.85
c) 1 – 0.85 = 0.15 (Use a Venn diagram)
PTS: 1
DIF: 3
OBJ: Section 6.5 LOC: A1.5
KEY: mutually exclusive events
33. ANS:
100% – (60% + 75% – 50%) = 15%
PTS: 1
LOC: A1.5
REF: Application | Communication
TOP: Counting and Probability
DIF: 2
REF: Application
TOP: Counting and Probability
4
OBJ: Section 6.5
KEY: mutually exclusive events
ID: A
PROBLEM
34. ANS:
a)
b) P(both red) =
4
C2 × 9 C0
13
C2
6
=
78
1
=
13
c) P(at least one green) = P(one green) + P(two greens)
ÊÁ C × C ˆ˜ + ÊÁ C × C ˆ˜
Ë6 1 7 1¯ Ë6 2 7 0¯
=
13 C 2
42 + 15
=
78
19
=
26
PTS: 1
DIF: 3
REF: Application
LOC: A2.5
TOP: Counting and Probability
KEY: probability using counting techniques
5
OBJ: Section 6.3
ID: A
35. ANS:
57
a) P(cold) =
179
=
0.3184
or about 31.8%
Ö
b) Using the conditional probability formula, P(A| B) =
P(A and B)
P(B)
:
47
P(healthy|female) =
179
97
179
47
=
97
=
0.4845
or about 48.5%
Ö
c) Restricting the sample space to only those who had the flu, P(male|flu) =
19
37
=
0.5135
or about 51.3%
Ö
d) Restricting the sample space to only males,
44
P(cold or flu| male) =
82
=
Ö 0.5366 or about 53.7%
PTS: 1
LOC: A1.6
36. ANS:
DIF: 3
REF: Application
TOP: Counting and Probability
The probability of rolling doubles on the first roll is
first roll is
5
6
OBJ: Section 6.4
KEY: dependent and independent events
6
1
= . The probability of not rolling doubles on the
36 6
.
Therefore, the probability of rolling doubles on the second roll is
5
6
1
×
The probability of rolling doubles on the first roll or the second roll is
6
1
5
=
36
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.
PTS: 1
LOC: A1.6
DIF: 4
REF: Thinking
TOP: Counting and Probability
6
+
=
OBJ: Section 6.4
KEY: dependent and independent events