Ch 6 and 7 Review
Matching
Match each of these terms with one of the phrases below.
a. sample space
e. outcomes
b. empirical probability
f. theoretical probability
c. complement
g. trials
d. subjective probability
h. event
____
____
1. the outcomes not included in a particular event
2. based upon intuition and previous experience
Short Answer
3. Explain the differences between the empirical probability, theoretical probability, and subjective probability.
4. Explain the meaning of the terms in the probability formula,
.
5. Two coins are tossed simultaneously. What is the probability of tossing a head and a tail?
6. A standard die is rolled. What is the probability of rolling a prime number?
7. Two standard dice are rolled. What is the probability that the total of the two dice is less than 4?
8. Two standard dice are rolled. What is the probability that a sum less than 7 is not rolled?
9. If the odds are 9:1 against the next car you see being red, what proportion of cars in your area are red?
10. Participants in marathons are often given numbers to wear, so that race officials can identify individual
runners more easily. If the numbers are assigned randomly, what is the probability that the eight fastest
runners will finish in the order of their assigned numbers, assuming that there are no ties?
11. 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.
12. If the probability of the Rangers defeating the Eagles in a hockey game is
, what is the probability that the
Rangers will win two consecutive games against the Eagles?
13. 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 publicspeaking contest?
c) What is the probability that she fails to be selected for either the students’ council or the public-speaking
contest?
14. What does a probability distribution show?
15. What is the key characteristic of a continuous variable?
16. What is the difference between a discrete random variable and a continuous random variable?
17. What is a uniform probability distribution?
18. Classify the following random variables as discrete or continuous.
a) the number of times a player bumps the ball in a volleyball game
b) the length of time a student spends doing homework
c) the volume of water in a swimming pool
d) the number of blue cars on a highway
19. Explain whether counting the number of red cars parked on a street is a probability experiment that has a
uniform probability distribution.
20. A die is rolled twenty times and the number of 1s is counted. Can this situation be modelled by a binomial
distribution? Justify your answer.
21. Explain why P(X = 2) = P(X = 3) in a binomial distribution where n = 5 and p = 0.5.
Problem
22. If a committee of six is to be chosen randomly from nine grade-11 students and seven grade-12 students, what
is the probability that the committee will be either all grade-12 students or all grade-11 students?
23. A survey of 50 female high-school athletes collected the following data.
Team
Field hockey
Volleyball
Rugby
Both rugby and field hockey
Both rugby and volleyball
Both field hockey and volleyball
All three teams
Number of Athletes
23
16
29
8
9
7
6
a) Draw a Venn diagram to illustrate the above data.
b) Determine the probability that a randomly selected athlete from this sample will play either rugby or field
hockey.
c) What is the probability that a randomly selected athlete will play on only one of the three sports teams?
d) Determine the probability that a randomly selected rugby player also plays volleyball.
e) Determine the probability that a randomly selected athlete who does not play rugby is on the field-hockey
team.
24. A spinner has three equally-sized sectors, numbered 1 through 3.
a) What is the probability that the arrow on the spinner will stop on an even number?
b) What is the expected outcome?
Ch 6 and 7 Review
Answer Section
MATCHING
1. ANS:
LOC:
2. ANS:
LOC:
C
CP2.06
D
CP2.06
REF:
TOP:
REF:
TOP:
Knowledge & Understanding
Probability concepts
Knowledge & Understanding
Probability concepts
OBJ: Section 6.1
OBJ: Section 6.1
SHORT ANSWER
3. ANS:
Empirical probability is based on experimental results and direct observations of events.
Theoretical probability is based on mathematical calculations and the analysis of the possible outcomes.
Subjective probability is based on past experience or informed guesswork.
REF: Communication
OBJ: Section 6.1
LOC: CP2.06
TOP: Probability concepts
4. 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).
REF: Communication
TOP: Probability concepts
5. ANS:
OBJ: Section 6.1
LOC: CP2.06
REF: Knowledge & Understanding
TOP: Calculating probability
6. ANS:
OBJ: Section 6.1
LOC: CP2.01
REF: Applications OBJ: Section 6.1
7. ANS:
LOC: CP2.01
TOP: Calculating probability
REF: Applications OBJ: Section 6.1
8. ANS:
LOC: CP2.01
TOP: Calculating probability
REF: Applications OBJ: Section 6.1
9. ANS:
10%
LOC: CP2.01
TOP: Calculating probability
REF: Applications OBJ: Section 6.2
10. ANS:
LOC: CP2.06
TOP: Calculating odds
REF: Applications OBJ: Section 6.3
11. ANS:
LOC: CP2.01
TOP: Calculating probability
The probability of picking the two red marbles in the first two picks is
.
Therefore, the odds in favour of picking the two red marbles are 1:44.
REF: Applications OBJ: Section 6.4
12. ANS:
LOC: CP2.01
TOP: Dependent and independent events
REF: Applications OBJ: Section 6.4
LOC: CP2.01
TOP: Dependent and independent events
13. ANS:
a) The two goals cannot be mutually exclusive since the probability of achieving both is 0.5.
b) 0.85
c) 0.15
REF: Applications, Communication
OBJ: Section 6.5
LOC: CP2.01
TOP: Mutually exclusive events
14. ANS:
A probability distribution shows the probabilities of all possible outcomes of an experiment.
REF: Knowledge & Understanding
OBJ: Section 7.1
LOC: CP2.03
TOP: Probability distribution concepts
15. ANS:
A continuous variable has an infinite number of different possible values in a continuous interval.
REF: Knowledge & Understanding
OBJ: Section 7.1
LOC: CP2.02
TOP: Probability distribution concepts
16. ANS:
A discrete random variable has separated values while a continuous random variable has an infinite number of
possible values along a continuous interval.
REF: Knowledge & Understanding
OBJ: Section 7.1
LOC: CP2.02
TOP: Probability distribution concepts
17. ANS:
A uniform probability distribution is a distribution in which all outcomes are equally likely in any single trial.
REF: Knowledge & Understanding
TOP: Probability distribution concepts
18. ANS:
a) discrete
b) continuous
c) continuous
d) discrete
OBJ: Section 7.1
LOC: CP2.03
REF: Knowledge & Understanding
OBJ: Section 7.1
LOC: CP2.02
TOP: Probability distribution concepts
19. ANS:
The experiment does not have a uniform probability distribution because all colours of cars are not equally
likely.
REF: Applications, Communication
OBJ: Section 7.1
LOC: CP2.03
TOP: Probability distribution concepts
20. ANS:
Yes, there are a given number of independent trials and rolling a 1 is a success.
REF: Communication
TOP: Binomial distribution concepts
21. ANS:
Since p = q and nCr = nCn–r, P(X = 2) =
REF:
TOP:
Applications, Communication
Binomial probabilities
OBJ: Section 7.2
LOC: CP2.05
= P(X = 3).
OBJ: Section 7.2
LOC: CP2.05
LOC: CP2.01
TOP: Mutually exclusive events
PROBLEM
22. ANS:
REF: Applications OBJ: Section 6.5
23. ANS:
a)
b) Only 6 athletes do not play either field hockey or rugby. Therefore, the probability of selecting an athlete
who plays either sport is
= 0.88.
c) From the Venn diagram, 38 athletes play on only one of the three sports teams. Therefore, P(only one
team) =
, or 0.76.
d) Venn Diagram Method
The Venn diagram above shows that 9 volleyball players also play on the rugby team. Therefore,
.
Conditional Probability Method
e) Venn Diagram Method
The Venn diagram above shows that 15 out of the 21 athletes not on the rugby team play field hockey.
Therefore,
Conditional Probability Method
REF: Applications, Communication
TOP: Mutually exclusive events
24. ANS:
OBJ: Section 6.5
LOC: CP2.01
a) The only even number on the spinner is 2, so the probability is .
b)
REF:
TOP:
Applications OBJ: Section 7.1
Expected value
LOC: CP2.03, CP2.04