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CHAPTER 13

1 Conducting Probability Experiments

Goal Compare probabilities in two experiments.

Game 1 Game 2 At-Home Help

1. Place a shuffled deck 1. Place a shuffled deck Probability refers to the likelihood
of cards face down. of cards face down. that an event will happen.
2. Turn over the top card. 2. Turn over the top card. For example, if you flip a coin, there
3. If the card is an ace, 3. If the card is a red card are two possible outcomes. You
you get 4 points. (heart or diamond), you can get either heads or tails.
get 2 points. The probability of getting heads is
A player wins if he or she
the same as the probability of
has at least 10 points A player wins if she or he getting tails.
after 4 turns. has at least 6 points after
4 turns.

1. Predict which game you are more likely to win. Justify your prediction.
Suggested answer: I think I’m more likely to get a red card than an ace. There are only 4 aces
in the deck, but there are 26 red cards. I think I’m more likely to win Game 2.

2. Tammy played both games three times.

Which game are you more likely to win? Use probability language to explain why.

Game 1 Points Game 2 Points

Turn number 1 2 3 4 Turn number 1 2 3 4

Ace?    ✓ 4 Red card? ✓ ✓  ✓ 6
Turn number 1 2 3 4 Turn number 1 2 3 4
Ace?   ✓  4 Red card?  ✓ ✓ ✓ 6
Turn number 1 2 3 4 Turn number 1 2 3 4
Ace?     0 Red card? ✓  ✓ ✓ 6

Suggested answer: You’re more likely to win Game 2. Tammy won Game 2 three times but lost
Game 1 every time. For Tammy to win Game 1, she would have to turn over 3 aces. Since there
are only 4 aces in the deck of 52, getting 3 of them would be very unlikely.
For Tammy to win Game 2, she would have to turn over 3 red cards. Since there are 26 red
cards in the deck of 52, it is more likely that Tammy can flip 3 of them to win.

116 Answers Chapter 13: Probability Copyright © 2006 Nelson

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CHAPTER 13

2 Using Percents to Describe Probabilities

Goal Conduct experiments and use percent to describe probabilities.

1. Siegfried rolled a die 20 times. At-Home Help

Roll 1 2 3 4 5 6 7 8 9 10 Probabilities can be written
as percents.
Number
2 3 1 4 2 6 3 5 2 1 For example, if you rolled a die
on die
10 times and got a 4 three times,
Roll 11 12 13 14 15 16 17 18 19 20
the probability of rolling a 4 would
Number 3
5 3 1 2 6 4 1 3 2 4 be 3 out of 10, or .
on die 10
3
To express  as a percent, find an
10
Record the probability of each event as a percent. equivalent fraction.
3 30
a) rolling a 1 c) rolling an odd number   
10 100
4 20 10 50  30%
 =   = 
20 100 20 100
= 20% = 50%
b) rolling a multiple of 3 d) rolling a 7
6 30 0 0
 =   = 
20 100 20 100
= 30% = 0%
2. The probability of winning a game is 40%. Predict how many times you expect
to win in each situation.

a) if you play 10 times b) if you play 25 times c) if you play 50 times

40 4 40 10 40 20
 =   =   = 
100 10 100 25 100 50
4 times 10 times 20 times
3. a) Roll a die 25 times and record each roll.
Suggested answer:
Roll 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number
on die 2 1 3 4 2 5 1 6 3 2 1 3 4 5 2 1 6 4 2 3 5 4 1 6 2

b) Record each probability as a percent.

i) rolling an even number ii) rolling a number iii) rolling a number less
less than 5 than 10
13 52 19 76 25 100
 =   =   = 
25 100 25 100 25 100
= 52% = 76% = 100%

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CHAPTER 13

3 Solving a Problem by
Conducting an Experiment
Goal Use an experiment as a problem solving strategy.

Jessica and her brother use a die to decide who will At-Home Help
help with dinner each night. An even number means To determine the probability in a
it is Jessica’s turn. An odd number means it is her problem, conduct an experiment.
brother’s turn. Conduct an experiment to determine All of the events in the experiment
the probability that Jessica will help with dinner more should be random. A result is
than 3 times in the next week. random if what happens is based
on chance. Something that is not
Suggested answer: random has to happen a certain way.

Understand the Problem For example, the day after Tuesday

I need to calculate the fraction of the days that Jessica will is always Wednesday. That is not
random. If you put the names of the
help with dinner in a week.
days of the week in a bag and pick
Make a Plan one name, the result is random.
I’ll conduct an experiment. I’ll roll a die 7 times. Each roll Remember to conduct many
represents a day of the week. I’ll record the results of experiments before determining
each roll in a chart. the probability in the problem.

I’ll do the experiment 20 times and see what fraction of the

days Jessica will help with dinner. If I get more than 3 even numbers in an experiment,
Jessica will help with dinner more than 3 times that week.
Carry Out the Plan
These are my results.
Experiment 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Roll 1 2 6 1 6 3 4 3 5 3 2 4 4 5 1 2 6 2 1 2 5
Roll 2 1 5 1 2 1 2 1 3 2 3 3 2 4 2 4 2 1 3 4 3
Roll 3 3 2 4 1 2 3 5 2 1 1 1 3 2 3 1 3 1 2 3 4
Roll 4 5 4 3 3 1 1 6 1 1 5 3 1 1 1 3 5 3 4 3 2
Roll 5 4 2 2 1 5 2 2 4 6 5 6 4 3 5 4 3 5 3 5 3
Roll 6 6 3 2 2 5 6 3 3 4 4 5 6 4 6 5 4 6 5 6 2
Roll 7 1 1 5 4 4 5 4 1 2 1 1 5 2 1 6 1 2 1 2 4
✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

I got more than 3 even numbers in 9 of the 20 experiments. So the probability of Jessica helping
9 45
with dinner more than 3 times in a week is . That is equivalent to  or 45%.
20 100
Look Back
There are 3 even numbers and 3 odd numbers on a die. So if I roll a die, there is a 3 in 6 chance
of getting an even number. If I roll a die 7 times, I expect to get an even number either 3 or 4 times.
My result of 45% looks reasonable
118 Answers Chapter 13: Probability Copyright © 2006 Nelson
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CHAPTER 13

4 Theoretical Probability
Goal Create a list of all possible outcomes to determine a probability.

1. If you shuffle a deck of cards, what is the theoretical At-Home Help

probability of each event? Theoretical probability is the
probability you would expect when
a) picking an ace c) picking a face card
you analyze all of the different
4 1 12 3
 or   or  possible outcomes.
52 13 52 13
For example, the theoretical
b) picking a spade probability of flipping a head on a coin
1
13 1 is , since there are 2 equally likely
 or  2
52 4 outcomes and only 1 is favourable.
Experimental probability is the
probability that actually happens
2. If you roll a die two times, what is the theoretical when you do the experiment.
probability of each event?
5 2 1
a) sum of 6  c) difference of 5  or 
36 36 18
Roll 1 Roll 1
1 2 3 4 5 6 1 2 3 4 5 6
Roll 2 1 2 3 4 5 6 7 Roll 2 1 0 1 2 3 4 5
2 3 4 5 6 7 8 2 1 0 1 2 3 4
3 4 5 6 7 8 9 3 2 1 0 1 2 3
4 5 6 7 8 9 10 4 3 2 1 0 1 2
5 6 7 8 9 10 11 5 4 3 2 1 0 1
6 7 8 9 10 11 12 6 5 4 3 2 1 0

b) sum of 10 3 1 d) difference of 2 8 2
 or   or 
36 12 36 9

1 2
Spin 1
3. Imagine spinning this spinner twice. 4 3 1 2 3 4
a) What is the theoretical probability Spin 2 1 2 3 4 5
that the sum of the two spins is 10 5 2 3 4 5 6
 or 
greater than 4? 16 8 3 4 5 6 7
b) What is the theoretical probability 4 5 6 7 8
8 1
 or 
that the sum is an odd number? 16 2

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CHAPTER 13

5 Tree Diagrams
Goal Use a tree diagram to determine a theoretical probability.

1. a) Use a tree diagram to list At-Home Help

2 3
the possible outcomes if A tree diagram is a way to record
this spinner is spun twice. 4 and count all combinations of events,
using lines to form branches.
Spin 1 Spin 2
2 For example, the tree diagram
2 3 below shows all the possible
4 outcomes if you flip a coin twice.
2
3 3 1st flip 2nd flip Outcome
4
H HH
2
4 3 H
4 T HT

b) Determine the theoretical probability H TH

that the difference of the numbers is 0. T
T TT
Tree diagram for parts b) and c):
Spin 1 Spin 2 Difference Product
2 0 4
2 3 1 6
4 2 8
2 1 6
3 1
3 3 0 9  or 
4 1 12 9 3
2 2 8
4 3 1 12
4 0 16

c) Determine the theoretical probability that the product of the numbers

is greater than 6.
6 2
 or 
9 3

2. Nathan and Jay are playing a game with the spinner in Question 1.
Nathan wins if his two spins give a sum greater than 5. Otherwise,
Jay wins. Use a tree diagram to explain if this game is fair. Spin 1 Spin 2 Sum
2 4
Suggested answer: The theoretical probability of getting a sum 2 3 5
6 2 4 6
greater than 5 is  or . A game is fair if each player has an
9 3 2 5
equal chance of winning. If there are two players, each player 3 3 6
should have a 50% chance of winning. In this game, Nathan is 4 7
2 6
more likely to get a sum greater than 5. So the game is not fair. 4 3 7
4 8

120 Answers Chapter 13: Probability Copyright © 2006 Nelson

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CHAPTER 13

6 Comparing Theoretical and

Experimental Probability
Compare the theoretical probability of an event with the results
Goal of an experiment.

1. Two green marbles, two blue marbles, and one At-Home Help
yellow marble are placed in a bag. The marbles To determine the theoretical
are mixed up and two marbles are picked, one probability of an event, you
at a time, without looking. can use a tree diagram to list
all possible outcomes.
a) What is the theoretical probability of picking
To determine the experimental
a green marble and then a yellow one? probability of that event, conduct
Use a tree diagram. an experiment.
Tree diagram for parts a) and b): Before comparing theoretical and
1st marble 2nd marble Experimental experimental probabilities, make
results sure the experiment was conducted
G2 1 many times.
G1 B1 2 Usually experimental probabilities
B2 2
are not the same as theoretical
Y 1
probabilities. If you do a great
G1 3
B1 1 enough number of experiments,
G2
B2 the experimental probability will
Y be the same as or very close to
G1 1 the theoretical one.
B1 G2 1
B2
Y 1
G1 1
B2 G2 1
B1 1
Y
G1 1
Y G2
B1 2
B2 1
2 1
 or 
20 10
b) Conduct an experiment 20 times. What is your experimental probability for this
event? Record your results beside your tree diagram in part a).
1
Suggested answer: 
20

c) Why might the experimental probability be different from the theoretical probability?
Suggested answer: The experimental probability was different because I only did
the experiment 20 times. Also, I might not have mixed up the marbles well enough
between experiments.

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CHAPTER 13

Test Yourself Page 1

Circle the correct answer.

Use the chart to answer Questions 1 and 2.
Nazir’s Rolls of a Die
First 5 rolls 2 1 6 3 5

Next 5 rolls 1 1 4 2 5

Next 5 rolls 3 2 4 5 1

1. What is the probability of Nazir rolling an even number in the first 10 rolls?
3 4 7 3
A.  B.  C.  D. 
10 10 10 5
2. What is the probability of Nazir rolling a number greater than 4 in all 15 rolls?
4 1 7 11
A.  B.  C.  D. 
15 3 15 15
3. What is the theoretical probability
of spinning blue on this spinner? red blue

1 2 yellow green
A.  C. 
6 3
blue purple
2 1
B.  D. 
6 2
4. Renata spun the spinner in Question 3 10 times. What is the probability of
Renata spinning blue?

Spin number 1 2 3 4 5 6 7 8 9 10

Colour blue yellow green red green green purple blue blue red

A. 10% B. 20% C. 30% D. 40%

5. What is the theoretical probability of flipping a coin three times and getting heads
all three times?
1 1 3 1
A.  B.  C.  D. 
8 4 8 2
6. What is the theoretical probability of picking an ace from a shuffled deck of cards?
1 4 1 3
A.  B.  C.  D. 
52 52 2 4

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CHAPTER 13

Test Yourself Page 2

7. One red counter, one blue counter, one green counter, and one yellow counter are
placed in a bag. The counters are mixed up and two counters are picked, one at a
time, without looking. Each time a counter is picked, it is not replaced in the bag.
Which tree diagram represents all possible outcomes?

A. 1st counter 2nd counter C. 1st counter 2nd counter

R B
R
B G
R
G
Y R
B
Y
R
B B
B G
G Y
Y
R
Y
R B
G B
G
Y
R
Y B
G
Y

B. 1st counter 2nd counter D. 1st counter 2nd counter

R B
R B R G
G Y

B R
B R B G
G Y

G R
G R G B
B Y

Y R
Y R Y B
B G

8. What is the theoretical probability of picking a green counter and a yellow counter
(in any order) for the situation in Question 7?
1 2 1 1
A.  B.  C.  D. 
12 12 4 3

Copyright © 2006 Nelson Answers Chapter 13: Probability 123