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CEGEP CHAMPLAIN - ST. LAWRENCE
201-510-LW: Business Statistics
Patrice Camiré
Problem Sheet #6
Basic Probability & Counting
1. In the context of a statistical experiment, define the following terms: outcome, sample space, and
event.
2. For each experiment, find the corresponding sample space S and #S.
(a) A die is rolled and the upper face is recorded.
(b) Two dice are rolled and the sum of their upper faces is recorded.
(c) Two dice are rolled and the difference of their upper faces is recorded.
(d) An urn contains a blue, a red, and a yellow ball. Two balls are randomly drawn (ordered and
without replacement).
(e) An urn contains a blue, a red, and two yellow balls. Three balls are randomly drawn (ordered
and without replacement).
(f) Two coins are tossed and the outcome is recorded (ordered).
(g) Two coins are tossed and the outcome is recorded (unordered).
3. State the Equiprobability Theorem.
4. A single die is rolled. Find the probability that the upper face is:
(a) equal to 6.
(c) even.
(b) not equal to 5.
(d) larger than 2.
5. A red die and a white die are rolled at the same time and their upper faces are observed. Find the
probability that:
(a) the upper faces are equal.
(b) the product of the upper faces is less than or equal to 4.
(c) the sum of the upper faces is larger than 7.
(d) the sum of the upper faces is less than or equal to 7.
6. A fair coin is flipped twice (ordered).
(a) Find the probability that only tails occurred.
(b) Find the probability of at least one heads.
7. A fair coin is flipped three times (ordered).
(a) Find the probability of obtaining heads exactly once.
(b) Find the probability of obtaining all heads.
(c) Find the probability of obtaining tails at least once.
8. A group of 100 business university students across the country was surveyed and the following results
were obtained:
20
52
25
13
15
17
10
read
read
read
read
read
read
read
Canadian Business;
the Ottawa Business Journal;
Maclean’s;
Canadian Business and the Ottawa Business Journal;
Canadian Business and Maclean’s;
the Ottawa Business Journal and Maclean’s;
all three magazines.
Using this data, find the probability that a randomly selected business university student will read:
(a) none of the three magazines.
(b) the Ottawa Business Journal only.
(c) Canadian Business and Maclean’s, but not the Ottawa Business Journal.
9. A survey of a group of 102 graduate students at McGill University showed that 9 of them have a
motorcycle, 22 have a car, 50 have a bicycle and 39 have neither a motorcycle, a car or a bicycle. No
one has at the same time a motorcycle and a car. What is the probability that a graduate student
chosen at random has a bicycle and either a car or a motorcycle?
10. Students in actuarial science at Concordia University can register in three specialized courses this
semester. It was found that 40% are attending the first course, 25% the second course and 15% the
third course. Moreover, 20% are attending two of the specialized courses and 5% are attending all
three. What is the percentage of students attending at least one of the three specialized courses?
Answers
1. An outcome is what is observed when performing an experiment.
The sample space is the collection/set of all possible outcomes.
An event is any collection/set of outcomes.
Remark: An outcome is also called a simple event.
2. (a) S = {1, 2, 3, 4, 5, 6} , #S = 6
(b) S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} , #S = 11
(c) S = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} , #S = 11
(d) S = {BR, BY, RB, RY, Y B, Y R} , #S = 6
(e) S = {BRY, BY R, BY Y, RBY, RY B, RY Y, Y BR, Y BY, Y RB, Y RY, Y Y B, Y Y R} , #S = 12
(f) S = {HH, HT, T H, T T } #S = 4
(g) S = {two heads, two tails, one head and one tail} , #S = 3
3. If the sample space S of a statistical experiment is finite and if all the outcomes are equally probable,
then for any event E ⊆ S,
#E
P (E) =
.
#S
1
6
1
5. (a)
6
1
6. (a)
4
3
7. (a)
8
4. (a)
8. (a) 0.38
9.
9
≈ 0.18
51
10. 50%
5
6
2
(b)
9
1
2
5
(c)
12
3
(b)
4
(b)
2
3
7
(d)
12
(c)
(b)
1
8
(b) 0.32
(d)
(c)
7
8
(c) 0.05