Basic Probability Concepts
1. Explain what each of the following probability values implies about the likelihood of event
A occurring.
a) P(A) = 1
b) P(A) = 0
c) P(A) = 0.5
2. What is the probability of drawing each of the following from a standard deck of cards?
a) a red card
b) a club
c) a 6
d) a face card
e) a black ace
3. A couple plans to have three children. Construct a tree diagram to illustrate the possible
gender outcomes of the three children.
4. Refer to question 3. What is the probability that the couple will have
a) three girls?
b) two boys?
c) not all girls?
d) a girl, then a boy, then a girl?
e) State any assumptions you must make in your calculations.
5. A coin is flipped ten times, and turns up heads six times.
a) Based on these results, what is the empirical probability of flipping heads with this
coin?
b) Can you conclude that this coin is unfair? Explain.
c) Explain how you can draw a better conclusion regarding the fairness of this coin.
6. Estimate subjective probabilities for each of the following events.
a) It will snow tomorrow.
b) The next school bell will ring on time.
c) Canada will win at least one gold medal in the next Olympic Games.
d) The price of gas will rise at least once in the next week.
7. For each of your estimates in question 6, provide a rationale for your answer.
8. What is the probability of not rolling an 11 with a standard pair of dice?
9. For what probability will an event be twice as likely to occur than not occur?
10. In a simple card game, Player A wins a point if either a face card, a red prime number, or a
black perfect square number is drawn from a standard deck. Otherwise, Player B wins a
point. Assume aces do not count as 1s.
a) Which player has the advantage in this game? Support your answer with calculations
and an explanation.
b) Modify the game to give the other player a slight advantage. Explain the new rules and
show that the other player now has the advantage.
c) Modify the game to make it fair to both players. Explain and show that each player has
an equal chance of winning.
Odds
1. What are the odds in favour of rolling each of the following totals with a standard pair of
dice?
a) 7
b) doubles
c) less than 4
d) greater than 4
2. Tomorrow’s forecast calls for a 40% chance of rain.
a) What are the odds in favour of a rainy day tomorrow?
b) What are the odds against a rainy day tomorrow?
3. What are the odds in favour of drawing the following from a standard deck of cards?
a) a 10
b) a spade
c) a black king
d) a red face card
4. Billie has 12 CDs in her travel pack, 5 of which are rock, 3 are dance, and 4 are country.
She reaches into her pack and randomly selects a CD.
a) What are the odds in favour of her selecting a rock CD?
b) What are the odds against her of selecting a dance CD?
5. In an effort to promote better driving habits among teens, a local ministry of transportation
office claims that the odds in favour of passing a drivers’ test on the first attempt is twice as
great for graduates of approved driver-training programs than for those who do not take
driver training. If the odds in favour of a driver-training graduate passing on his or her first
attempt is 2:3,
a) what are the odds in favour of a non-graduate passing the first test?
b) what are the odds against a non-graduate passing the first test?
c) what is the probability that a graduate will pass the first driver test?
d) what is the probability that a non-graduate will pass the first test?
6. A sports commentator gives the home team 2:1 odds in favour of winning a hockey game,
and estimates a 10% probability of a tie. Based on these estimates, how likely is the home
team to lose?
Probability Using Counting Techniques
1. A pencil case contains four pencils, three blue pens, and two red pens. If two writing instruments
are randomly selected, what is the probability that they will be
a) two pencils?
b) two pens?
c) two different coloured pens?
d) a blue pen followed by a red pen?
2. A family of two parents and three children lines up in a row for a photograph. If the three
children are randomly placed in a row in front of their parents, what is the probability that they
will be arranged in ascending or descending order of age?
3. A committee of four members is to be randomly formed from eight technicians, four engineers, and
three mathematicians. What is the probability that the committee
a) will be comprised of all technicians?
b) will not be comprised of technicians only?
c) will be comprised of two technicians, one engineer, and one mathematician?
4. A bag of fast-food sandwiches contains two hamburgers, four cheeseburgers, and three
chicken burgers. If Wyatt ordered a cheeseburger and a chicken burger, what is the
probability that
a) the first sandwich selected will be one of his choices?
b) neither of the first two sandwiches will be among Wyatt’s choices?
5. Petra has mastered the proofs for six of ten theorems. Two of the theorems will appear as
questions on the final examination. What is the probability that
a) neither of the proofs on the examination will be among those Petra mastered?
b) at least one of the mastered proofs will appear on the examination?
6. A pizza contains three toppings randomly selected from the following choices: pepperoni,
mushrooms, green peppers, ham, bacon, and onions. Manuel enjoys all toppings, except bacon and
onions. What is the likelihood that a pizza will contain
a) no toppings that Manuel dislikes?
b) exactly one topping that Manuel dislikes?
c) at least one topping that Manuel dislikes?
7. A CD club has a special offer in which each member will receive two of the top ten selling CDs
from last year, one in each of the next two monthly shipments.
a) What is the probability that a member will receive two of the top five selling CDs?
b) What is the probability that a member will receive the number one selling CD in the
first month, followed by the number two selling CD?
8. A volleyball team consists of nine players, six of whom are on the court at any given time. If the
starting line-up is randomly chosen, what is the probability that
a) the captain and both alternate captains will start the game?
b) none of the three team leaders will start the game?
Answers
Probability Using Counting Techniques
Basic Probability Concepts
1.
2.
a) certainty
b) impossible
c) 50% likely
1
1
1
a)
b)
c)
2
13
4
3
1
d)
e)
26
13
3.
1.
c)
2.
G
B
G
B
G
B
G
B
G
4.
5.
6.
7.
8.
9.
a)
1
b)
B
3
c)
7
d)
18
2
3
7
> 0.5
13
b), c) Answers may vary.
Odds
1.
2.
3.
4.
5.
6.
1
8
8
8
8
e) Answers may vary, for example, no twins or triplets
on third pregnancy.
a) 0.6
b) No, sample size is too small.
c) Conduct many more trials.
a) Answers may vary.
b) Answers may vary, for example, 0.9.
c) Answers may vary, for example, 0.9.
d) Answers may vary.
Answers may vary.
17
10. a) B; P(B) =
a) 1:5 b) 1:5
c) 1:11 d) 5:1
a) 2:3 b) 3:2
a) 1:12 b) 1:3
c) 1:25 d) 3:23
a) 5:7 b) 3:1
a) 1:3 b) 3:1
c) 0.4 d) 0.25
23%
6
1
b)
d)
5
18
1
12
1
3
B
B
1
6
G
G
a)
3.
a)
4.
a)
5.
a)
6.
a)
7.
a)
8.
a)
2
39
7
9
2
15
1
5
2
9
5
12
b)
b)
b)
b)
b)
b)
37
c)
39
1
36
13
15
3
5
1
90
1
84
c)
4
5
16
65