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Key Words
What
You’ll Learn
To use mathematics to
describe and analyse the
probability of events
• probability
• event
And Why
Understanding probability
can help us make sense of
real-life situations that involve
uncertainty and randomness.
• theoretical probability
• experimental
•
•
•
•
probability
frequency
frequency distribution
trial
simulation
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CHAPTER
8
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Activate Prior Knowledge
Probability Vocabulary
Prior Knowledge for 8.1
In weather forecasts, the likelihood or chance of getting rain can be given as a probability.
The weather forecast is never certain.
However, some weather conditions indicate that it is more likely to rain than not rain.
Example
Tamara put 6 congruent cubes in a hat: 4 green, 1 red, and 1 blue.
Without looking, she’ll take a cube out of the hat.
What is the likelihood that Tamara will get each of these?
a) A blue cube
b) A green cube
c) A cylinder
d) A cube
Solution
There are 6 cubes in the hat, so there are 6 possible outcomes.
a) Only 1 of the 6 cubes is blue. It is unlikely that Tamara will get a blue cube.
b) Four of the 6 cubes are green. It is likely that Tamara will get a green cube.
c) There are no cylinders. It is impossible that Tamara will get a cylinder.
d) All of the objects in the hat are cubes. It is certain that Tamara will get a cube.
✓ Check
1. Use the words impossible, unlikely, likely, and certain to describe each event.
Tomorrow will be sunny.
b) The Toronto Raptors will win the Stanley Cup this year.
c) Someone in your class is 16 years old.
d) Ten coins flipped will all show heads.
a)
2. Ron and Jon will each spin this spinner once.
Order these events from least likely to most likely:
A. Both spin red.
B. Both spin green.
C.
At least one spins green.
3. Describe an event in your life for each likelihood. Explain your thinking.
a)
346
unlikely
b)
CHAPTER 8: Applying Probability
impossible
c)
certain
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Fractions, Decimals, and Percents
Prior Knowledge for 8.1
Fractions, decimals, and percents are ways to express parts of a whole.
In the diagram, the shaded part can be expressed as a fraction: 25
To express a fraction as a decimal, divide the numerator by the
denominator:
2 5 = 0.4
To express a decimal as a percent, write it as a fraction with hundredths:
4
40
0.4 = 10
= 100
= 40%
So, 25 = 0.4 = 40%
Example
A hockey goalie faced 28 shots and made 24 saves in a game.
a) Express the saves as a fraction of the shots in lowest terms, or simplest form.
b) Use a decimal with 3 decimal places to express the number of saves per shot faced.
c) Determine the goalie’s save percent in this game.
Solution
Divide the numerator and denominator by
= 24 4 = 67
the greatest common factor 4.
28 4
b) 6 7 = 0.85714 . . .
6 7 = 0.857 to 3 decimal places
The number of saves per shot faced is about 0.857.
c) 0.857 = 85.7%
To write a decimal as a
percent, multiply by 100.
The goalie’s save percent is about 85.7%.
a)
24
28
✓ Check
1. Express each fraction as a decimal and as a percent.
a)
6
10
b)
3
27
c)
9
17
d)
24
15
2. A football quarterback attempted 27 passes in a game and had 15 completions.
Express the completions as a fraction of the passes.
b) Express the fraction in part a as a percent. It is the quarterback’s completion percent.
a)
3.
Describe a situation when you would express a number in each form. Explain your
choice.
a) Fraction
b) Decimal
c) Percent
Activate Prior Knowledge
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Making Strong Notes
Within 24 h, people forget as much as 80% of the information they hear or
read. Creating strong notes can help you remember what you have learned.
To make strong notes about a concept you have learned, you need to
prioritize, condense, and organize the important information about the topic.
Step 1
Step 2
➢ Assign strength ratings.
➢ Give main ideas or categories
a strength rating of 1.
➢ Give supporting details and
examples strength ratings of
2, 3, or 4.
➢ Use indents, bullets, and
colour to organize your
strong notes.
➢ This will help you see how
the concepts relate.
Main idea
Strength rating 1
Data Management
Brief explanation
Strength rating 3
Measures of Central Tendency
Mean
Sum of a set of numbers, divided by the number of numbers:
14, 18, 17, 16, 16, 15; the mean is 96 = 16
6
Further details
about the first
Median
main idea
The middle number when data are arranged in order,
Strength rating 2
or the mean of the two middle numbers:
14, 15, 16, 16, 17, 18; the median is 16 + 16 = 16
2
Mode
The number that occurs most often in a set:
14, 15, 16, 16, 17, 18; the mode is 16
Example
Strength rating 4
Make strong notes as you study probability.
When you study, cover up portions of your notes.
Practise filling in the missing information.
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8.1
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Probability in Our Lives
Graduates of college advertising programs often begin their careers as media planners.
A media planner works with a client to determine a target audience.
She then buys television, radio, and print advertising to best reach this audience.
To do this, a media planner makes decisions based on statistical data.
Investigate
Using Probability to Make Decisions
Work with a partner to answer each question.
Explain your answers and identify the statistic or probability
that helped you make your decision.
AT ONTARIO
T REPORTS TH
GOVERNMEN
ONE-THIRD
TO GROW BY
POPULATION
XT 25 YEARS!
OVER THE NE
ide money
would you prov
As an investor,
n?
using subdivisio
to build a new ho
HOSPITAL LOTTERY:
1 in 7 tickets wins
Would you buy a ticket?
91% of Canadian
drivers buckle up
!
Are roadside seatb
elt checks
a good use of polic
e time and efforts?
60% CH
ANCE O
F RAIN
TOMOR
ROW
Would y
ou canc
el your p
lans
to go for
a hike?
v%)
save percent (S
An NHL goalie’s
s he
ot
e number of sh
is the ratio of th
ces.
fa
he
mber of shots
stops to the nu
:
on
6 seas
For the 2005/200
% was 0.929
Sv
’s
Cristobal Huet
0
’s Sv% was 0.88
Jussi Markkanen
ore
more likely to sc
Would a player be
Markkanen?
against Huet or
Reflect
➢ Is there other information that might have helped you answer
some of these questions? If so, describe it and explain how it
might affect your answer.
➢ Compare your answers with those of another pair.
If you disagree, what is the reason?
➢ These data were presented using ratios, percents, fractions, and
decimals. Which did you find easiest to interpret? Explain.
8.1 Probability in Our Lives
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Connect the Ideas
Probability is a way of describing chance or likelihood.
Probabilities can be determined from experimental results,
by analysing possible outcomes, or by interpreting statistical data.
Ways to describe
likelihood
We often express likelihood using words such as certain, likely, unlikely,
and impossible. We can also use numbers.
An event that is impossible has probability 0.
An event that is certain has probability 1.
We can express all other probabilities using decimals between 0 and 1.
For example, an event that is just as likely to occur as not occur
has probability 0.5.
Probability lines
Very unlikely
Impossible 0.10
0
Unlikely
0.20
0.30
Likely
0.40
0.50
0.60
Very likely
0.70
0.80
0.90 Certain
1
We can also use fractions and percents to express probability.
Very unlikely
Impossible
0% or 0
Connecting
statistics and
probability
350
20% or
Unlikely
Likely
50% or
Very likely
80% or
Certain
100% or 1
Sometimes, we use statistics to generate probabilities.
For example, by March 2006, 52% of Canadians had cell phones.
In Sweden, more than 90% of the population had cell phones.
Ottawa and Stockholm, the capitals of the countries, have similar
populations.
Suppose you were walking on a street in Ottawa or Stockholm.
In which city are you likely to see more people using cell phones?
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Since the populations of the cities are similar, you might expect
to pass about the same number of people as you walk in each city.
Since 52% of Canadians have cell phones, then about one-half
of the people you pass might have a cell phone.
Since more than 90% of Swedes have cell phones, then
most of the people you pass might have a cell phone.
Very unlikely
Unlikely
Very likely
90%
Likely
52%
Impossible 10%
0
20%
30%
40%
50%
60%
70%
80%
90% Certain
100%
So, you are likely to see more people using cell phones in Stockholm.
Practice
1. Use probability vocabulary to describe the likelihood of each event.
a)
b)
c)
d)
e)
You will meet someone new today.
The temperature will be greater than 25°C tomorrow.
You will have a test tomorrow.
The next person you see will be famous.
You will write something in the next 5 min.
2. Write a decimal that could be used to express the probability of each event in question 1.
Use a probability line to help order the events from least to most likely.
3. a) Is a coin toss a good way to determine which team will take the opening kick
in a soccer game? Explain why or why not.
b) Describe a decision in your life you made or might make using a coin toss.
4. If a basketball player is fouled while shooting, he or she is awarded
at least one free throw.
Midway through the 2006–2007 season, Shaquille O’Neal’s career
free throw average was 0.528.
Steve Nash’s career free throw average was 0.895.
Which player do you think is more likely to score a basket
on his next free throw? Explain your answer.
A free throw average
is the number of
baskets made on free
throws divided by the
number of attempts.
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5. a) Use newspapers or magazines to find three examples of probability
You will need
or statistics that could be used to write a probability.
newspapers or
magazines.
Is the information presented as a fraction, percent, decimal, or ratio?
What does the probability or statistic mean?
b) For each example in part a, write a question someone could answer using the data.
You can use survey results to make predictions about the population represented in the survey.
Example
Choosing someone at
random from a group
means that each
person in the group
has the same chance
of being selected.
Solution
In a survey, 140 out of 350 people report that they have made online purchases
using the Internet.
a) What is the probability that someone chosen at random from the survey
respondents has made an online purchase?
Express your answer as a fraction, percent, decimal.
Show your answer on a probability line.
b) Suppose this survey is representative of the general population.
If you asked 1000 people at random, how many might you expect
to have made online purchases?
a)
Of the 350 respondents, 140 have made online purchases.
So, if you choose a respondent at random, the probability of choosing one
who has made an online purchase is 140 out of 350, or 140
70
350
You can express this fraction in lowest terms
by dividing the numerator and denominator
by the greatest common factor, 70
140
350
=
2
5
70
To express this fraction as a decimal,
divide the numerator by the denominator:
140 350, or 2 5 = 0.4
To express this decimal as a percent, multiply by 100%:
0.4 100% = 40%
The probability that a survey respondent chosen at random
has made a purchase an online purchase is 25 , 0.4, or 40%.
140 people
No people
0
b)
352
10%
20%
30%
40% 50%
or 0.4
350 people
60%
70%
80%
90% 100%
For a sample of 1000 people, use the probability in decimal form.
When you select people randomly, you would expect 0.4 1000, or
400 of the people you ask to have made online purchases.
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6. According to Statistics Canada, in 2005, 64% of Canadians 12 and older lived
in smoke-free households.
a) What is the probability that a Canadian 12 or older selected at random lives
in a smoke-free household? Express your answer as a fraction, decimal, and percent.
b) How many people in a group of 500 Canadians 12 or older selected
Assume that the
at random might you expect to live in a smoke-free household?
data from 2005
are still valid.
Explain the strategy you used to determine your answer.
How did you decide whether to work with the probability expressed
as a fraction, decimal, or percent?
7. Assessment Focus The table shows how many
Audience
Canadians watched two television shows in one week.
Age group
Show A
a) Which of the two shows is a 23-year old selected at
Under 18
220 000
random more likely to have watched?
18 to 24
450 000
How do you know?
25 to 34
230 000
b) Which show had more viewers?
35 to 49
45 000
Explain how you determined your answer.
50 to 54
15 000
c) Suppose a person watching Show B is selected
55+
3000
at random. What is the probability that the person
is in the 50 to 54 age group?
d) Suppose you are a media planner buying advertising time for a new mp3 player.
During which show would you rather place an ad?
Explain your reasons.
e) Would your answer to part d change if the product you are advertising
is a video game? Explain your answer.
Show B
5000
150 000
325 000
460 000
400 000
275 000
8. Take It Further This quote appeared in a newspaper article in 2007:
“Nearly 70 per cent of all new jobs will require some level of post-secondary education.
But only 53 per cent of Canadians graduate from college or university . . . .”
Based on these data, will there be enough Canadians with the skills required to fill the new jobs
being created? Use probability to help explain your answer.
Think about events in your own life that are certain to happen, likely
to happen, or impossible. Describe a real-life event that is an example
of each likelihood. Justify your reasoning.
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Even-Odd Football
Materials
• 2 dice
• a small object to use as a counter
• Even-Odd Football game board
Even Goal
Starting Line
Play with a partner.
➢ Decide who will aim for the even goal
and who will aim for the odd goal.
➢ Place the counter on the starting line.
The first player rolls the dice.
Try this game after
you have completed
Section 8.2.
Odd Goal
➢ Subtract the lesser number from the greater number.
• If the difference is odd, move the counter one space
toward the odd goal.
• If the difference is even, move the counter one space
toward the even goal.
• If the difference is zero, do not move the counter.
➢ Take turns rolling the dice and moving the counter until it reaches
a goal. The player who was aiming for that goal wins.
After you play the game several times, answer these questions.
A fair game is one in
which each player
has an equal chance
of winning.
1. Do you think this game is fair? If not, would you rather be aiming
for the odd or even goal? Explain your thoughts.
2. a) Draw a table to show the differences
First die
when two dice are rolled.
1 2 3 4 5 6
b) Use the table to determine the
1
probability of the difference being
2
odd, the probability of the difference
3
being even, and the probability
4
of the difference being 0.
5
c) Now do you think the game is fair
6
or unfair? Explain your thinking.
d) If you think the game is unfair,
suggest a way to make it fair. Try out your idea.
Second die
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Theoretical Probability
Computer programmers have created reservation systems for theatres, airplanes,
and concert halls that automatically assign the best available seats to ticket buyers.
Suppose seats were assigned randomly instead. Do you think people would be happy
with their seats?
Investigate
Comparing the Likelihood of Events
The seats in a small concert hall are arranged in 10 rows with 10 seats
in each row. The 4 seats in the middle of each row are separated
from the 3 seats on the far left and the 3 seats on the far right by aisles.
➢ How many seats are in the hall?
➢ How many seats of each type listed below are there?
How do you know?
• seats in the front row
• aisle seats
• seats in the last 6 rows
Row
10
9
8
7
6
5
4
3
2
1
Stage
Suppose you are the first person to buy a ticket.
You are randomly assigned a seat.
➢ Which type of seat listed above are you most likely to get?
Order the possibilities from most to least likely.
Explain your strategy for ordering them.
➢ Which seat assignment is more likely, or are they equally likely?
Explain your thinking.
• You are assigned one of the two best seats in the theatre,
in the middle of the 6th row.
• You are assigned one of the worst seats in the theatre,
the 3 seats near the wall at either end of the front row.
Reflect
➢ How could you write a fraction, decimal, or percent to describe
the likelihood of receiving each type of seat described above?
➢ In real life, seats in a concert hall are not assigned randomly.
What influences the seat you choose in a concert hall?
Which seats do you think are most likely to be chosen?
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Connect the Ideas
Theoretical probability is the number assigned to
the likelihood of an event happening.
It is determined by analyzing all the possible outcomes.
It is often simply called probability.
A card is drawn at random from a standard deck of
playing cards.
What is the theoretical probability of drawing a king?
There are 52 cards in a standard deck.
Each card is equally likely to be drawn.
So, there are 52 equally likely outcomes
when 1 card is drawn.
Four cards are kings: K♠, K♥, K♦, K♣
5
So, the probability of drawing a king is: 4 in 52, or 52
In lowest terms
It can be helpful to express the probability in different forms.
Divide the numerator and denominator
4
by the greatest common factor, 4.
1
4
= 13
52
4
As a decimal
As a percent
0.08
Divide the numerator by the denominator: 4 52 or 1 13 Multiply the decimal by 100%:
0.08 100% = 8%
1
The probability that the card will be a king is 13
, or about 0.08,
or about 8%.
On a probability line, this lies just to the left of 0.1.
Very unlikely
Impossible 0.10
0
0.20
Unlikely
0.30
Likely
0.40
0.50
0.60
Very likely
0.70
0.80
0.90 Certain
1
As you can see from the line, a probability of 0.08 means
you are very unlikely to draw a king when you draw one card.
You may draw a king. But generally, you would have to draw and
replace several cards before you get a king.
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When you calculate the theoretical probability of an event, you must:
• count the possible outcomes;
• know that the outcomes are equally likely; and
• count the outcomes that are favourable to the event.
You can then use this formula.
outcomes favourable to the event
Probability of an event = Number of
Total number of outcomes
Practice
1. A regular die is a cube with faces labelled 1 to 6. The die is rolled.
What are all the possible outcomes?
Are they equally likely? Explain how you know.
b) What is the probability of rolling a 2?
c) What is the probability of rolling an odd number?
d) What is the probability of rolling a 7?
a)
2. The pointer on this spinner is spun.
What are all of the possible outcomes?
Are they equally likely? Explain how you know.
b) What is the probability of spinning 1?
c) What is the probability of spinning an even number?
d) What is the probability of spinning either 1, 2, or 3?
a)
4
1
3
2
3. Three spinners are shown.
Suppose you want to spin blue.
Without calculating any probabilities, explain which spinner you would use.
b) Suppose you do not want to spin blue.
Without calculating any probabilities, explain which spinner you would use.
c) Calculate the theoretical probability of spinning blue on each spinner.
Did you make the best choices in parts a and b? Explain your thinking.
a)
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4. Copy and complete this table to show all the possible sums when two dice are rolled.
Determine the probability of each event.
a) Rolling a sum of 12
b) Rolling a sum of 7
c) Rolling a sum greater than 7
d) Rolling a sum less than 6
Sum When Two Dice are Rolled
First die
1
2
3
4
5
6
1
Second die
Chapter 08
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3
4
5
6
5. Assessment Focus Lucky Guess is a game that can be used for fundraising.
Pictures of prizes are hidden behind some of the squares on a 5-by-5 game board.
Players pay for the chance to uncover a square.
If a prize is shown, the player wins it.
Ten prizes are hidden on the game board.
Suppose you are the first person to pick a square.
What is the probability you will win a prize? Express your answer as a fraction.
b) Suppose the conditions are the same as described in part a. What is the probability
that you will not win a prize? Explain how you determined your answer.
c) How many prizes should the game organizer hide so the probability
of the first player winning a prize is 15 ? Explain your method.
a)
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A tree diagram can help you to list all possible outcomes of a probability situation.
Example
A perfect square is
a number that is the
square of a whole
number.
Contestants in a carnival game spin pointers on three spinners. The numbers
that the first and third pointers land on are either added or multiplied, as
determined by the second spinner. To win a prize, the result must be a perfect
square. For each attempt, what is the theoretical probability of winning a prize?
+
2
4
2
4
×
Solution
Draw a tree diagram to show all the possible outcomes.
Outcomes
of first
spinner
Outcomes
of second
spinner
+
2
×
+
4
×
Outcomes
of third
spinner
Possible sums
and products
2
2+2=4
4
2+4=6
2
2×2=4
4
2×4=8
2
4+2=6
4
4+4=8
2
4×2=8
4
4 × 4 = 16
There are 8 possible outcomes. All are equally likely.
The only perfect squares are 4 and 16.
There are two ways to obtain a 4, and one way to obtain 16.
So, the probability of winning a prize is 38 , or 0.375, or 37.5%.
6. Suppose you toss a coin three times.
Which event do you think is more likely: you get 3 heads
or you get 1 head and 2 tails? Explain your thinking.
b) Draw a tree diagram to show the possible outcomes when a coin
is tossed three times. Are the outcomes equally likely?
c) Use your diagram to determine the probability of each event. Explain your methods.
i) 3 heads
ii) no heads
iii) 1 head and 2 tails
d) Was your answer to part a correct? How do you know?
a)
8.2 Theoretical Probability
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7. Two students invent a game. Two players race pieces around a
game board. They roll a die to determine how many spaces to
Both
Your
move. They use a spinner to determine whether the player moves
pieces
piece
her or his own piece, the opponent’s piece, or both pieces.
a) Make a table or draw a tree diagram to show the possible
Your
opponent’s
outcomes when the die is rolled and the spinner spun. Explain
piece
the reasons for your choice.
Are the outcomes equally likely?
b) Suppose you are playing the game. You roll the die and spin the pointer.
Determine the probability of each event.
i) You move only your own piece 6 spaces.
ii) You move either or both pieces 3 spaces.
iii) You move only your opponent’s piece any number of spaces.
8. A vending machine at a video store contains different colours of gumballs.
When you put in a coin and turn the handle, one gumball drops out.
If the ball is white, you win a free movie rental.
When Alyssa puts in her coin, the machine contains 10 pink gumballs,
11 blue gumballs, 8 green gumballs, 14 yellow gumballs, 15 purple gumballs,
and 2 white gumballs.
a) What is the theoretical probability that she gets a pink gumball?
b) What is the theoretical probability that she gets a blue or yellow gumball?
c) What is the theoretical probability that she wins a free movie rental?
d) Which colour of gumball does Alyssa have a 1 in 4 chance of getting?
Explain how you determined your answer.
9. Take it Further Thom had 7 coins in his hand with a total value of $2.39.
While waiting in line at the school cafeteria, he dropped one coin that rolled
under the counter. What is the probability that the coin he dropped was a penny?
Explain your strategy.
Describe the steps you should follow to calculate a theoretical
probability. Include a sample calculation involving coins, dice,
or spinners in your explanation.
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Mid-Chapter Review
8.1 1. Think of at least three words that have
been used in this unit to describe the
likelihood of an event.
Draw a probability line from 0 to 1. Write
each word in an appropriate place on the
line.
2. A survey reports, “Canadians aged 15 to
24 have a higher rate of volunteering
(55%) than any other age group.”
a) What is the probability that a 15-to
24-year-old selected at random does
volunteer work?
b) In a group of twenty 15- to 24-yearolds selected at random, how many
might you expect to do volunteer
work?
8.2 3. Zoe shuffles a set of cards numbered
1 to 10 and randomly picks a card.
a) What are the possible outcomes? Are
they equally likely? Explain how you
know.
b) What is the probability that the card
Zoe picks shows:
i) an even number
ii) a perfect square
iii) a two digit number
iv) a three digit number
3
4
2
6
well-shuffled deck of playing cards.
a) Without calculating the probabilities,
order these events from least likely to
most likely. Explain your reasoning.
• The card is a club.
• The card is red.
• The card is an ace.
b) Calculate the probability of each event
in part a. Express each answer as a
decimal.
c) Draw a probability line. Place your
answers to part b on the line.
How accurate was your ordering
in part a?
5. Contestants in a carnival game spin
pointers on two spinners.
The numbers the pointers land on are
multiplied. To win a prize, the product
must be an odd number.
1
2
4
3
5
6
Make a table or draw a tree diagram
to list all the possible outcomes.
Explain your choice.
b) For each attempt, what is the
probability of winning a prize?
a)
1
5
9
8
4. Misha randomly draws a card from a
7
10
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Experimental Probability
When a probability is calculated from the results of an experiment or survey, it is called an
experimental probability.
Some games, such as poker and bridge, involve making decisions based on probabilities. The
more you play these games, the better you should get. Without even realizing it, you will use your
observations from previous games to help you make good decisions.
Investigate
Comparing Theoretical and Experimental Probability
Some students are creating a board game. They think about making
a rule that you can only start moving your piece when you roll a 6.
➢ What is the theoretical probability of rolling a 6 with a die?
Express your answer as a fraction and a decimal, to the nearest
hundredth. Explain how you determined the probability.
➢ Suppose you were to roll a die 50 times, how many times
do you think you would roll a 6? Explain your thinking.
Work with a partner. You will need a die. If you have several dice, you
can speed up the experiment by rolling them at the same time.
➢ Roll a die. Record the outcome by adding a tally mark to a table like
this. Continue to get the data for 50 rolls.
This is called a frequency
distribution table.
Outcome
Tally
Frequency
1
2
3
4
5
6
➢ Record the frequency, which is the number
of rolls for each outcome.
Express your experimental
probabilities as fractions
and decimals to the
nearest hundredth.
362
➢ What fraction of the outcomes are 6?
This is the experimental probability of
rolling a 6.
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➢ Share your data with those of another pair.
Of the 100 rolls, how many rolls were 6s?
What is the experimental probability of rolling a 6 in 100 rolls?
➢ Share your data with other pairs until you have data for the entire
the class.
For how many rolls do you now have data?
How many rolls were 6s?
What is the experimental probability of rolling a 6?
➢ Graph the data for rolling a 6. Describe the shape of the graph.
➢ Draw a line on the graph to show the theoretical probability
of rolling a 6.
Experimental Probability of Rolling a 6
0.8
0.6
Probability
Chapter 08
0.4
0.2
0
100
200
Number of rolls
300
400
Reflect
➢ How did the theoretical and experimental probabilities compare?
Does this surprise you? Why or why not?
➢ Suppose you could collect data for 1000 rolls.
How do you think the theoretical and experimental probabilities
would compare? Explain your thinking.
➢ How likely is a person to roll a 6?
Do you think the students’ rule about needing to roll a 6
is a good one? Why or why not?
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Connect the Ideas
Each repetition of a probability experiment is called a trial.
In Investigate, you collected data for 50 rolls.
So, you conducted 50 trials.
Once you have collected data, you can calculate the experimental
probability of an event.
of times the event happens
Experimental probability = NumberTotal
number of trials
Conduct the
experiment
For example, Erika tossed a coin 50 times and recorded her results
in a frequency distribution table.
Coin Toss Experiment
Outcome
Tally
Frequency
Heads
|||| |||| |||| |||| ||
22
Tails
|||| |||| |||| |||| |||| |||
28
In Erika’s experiment, the frequency of heads is 22.
The frequency of tails is 28.
So, for Erika’s experiment:
Determine the
frequency and
experimental
probability
Number of heads
Experimental probability of Heads = Total
number of tosses
= 22
50
= 0.44, or 44%
Experimental probability of Tails
Number of tails
= Total
number of tosses
= 28
50
= 0.56, or 56%
Compare to the
theoretical
probability, if it
can be calculated
364
When tossing a coin, there are two possible, equally likely outcomes:
a head and a tail
So, the theoretical probability of tossing a head is 12 , or 0.5, or 50%.
The theoretical probability of tossing a tail is also 12 , or 0.5, or 50%.
Erika’s results are close to the theoretical probabilities.
This is not always the case.
The theoretical probability only tells us that on many repetitions,
you may expect to toss heads about half the time.
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Practice
1. A softball player’s batting average is the experimental probability:
the number of hits divided by the number of at bats.
Batting statistics for part of the 2006 season are shown for 3 players from the University of
Regina Wildcats women’s softball team.
Name
Times at bat
Hits
a) Calculate each player’s batting average.
b) Which player is most likely to get a hit
Kendra Dayle
45
15
her next time at bat? Explain your
Amery Deren
40
10
reasoning.
Lauren McDonald
26
9
c) Why can you not say for certain who
will get a hit and who will not?
2. a) Use your data from Investigate.
b)
c)
d)
e)
Calculate the experimental probability of rolling an odd number.
What is the theoretical probability of rolling an odd number?
How do your answers to parts a and b compare?
Suppose you collected data for another 50 rolls. How do you think
the experimental probability might change? Explain your thinking.
Share data with another student who was not your partner in Investigate.
Calculate the experimental probability for 100 rolls.
Did what you predicted in part d happen? Explain.
3. Assessment Focus There are four suits in a deck of 52 playing cards:
You will need a
deck of playing
cards.
hearts, diamonds, clubs, and spades.
a) Suppose you draw a card at random from a shuffled deck.
What is the theoretical probability of drawing a heart? Explain how you know.
b) Make a frequency distribution table.
Suit
Tally
Frequency
Get a deck of cards and shuffle them.
Hearts
Draw a card, record the suit in the table, and
Diamonds
replace the card. Repeat for 40
Clubs
trials altogether.
Spades
c) Calculate the experimental probability of drawing
a heart. Compare it to your answer to part a.
d) Suppose you could get data for 80 more trials.
How do you think the experimental probability might change? Explain.
e) Share data with 2 other students.
Calculate the experimental probability for 120 trials.
Did what you predicted in part d happen? Explain.
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4. Lia is performing a coin toss experiment. She gets 1 tail, then 4 heads.
Do you think she is more likely to get tails or heads on the next toss,
or are they equally likely? Use probability language to explain your reasoning.
A simulation is an experiment that models a real-life situation.
Example
Suppose a traffic light is programmed to be green 75% of the time
and red or yellow 25% of the time for cars travelling east and west.
a) What is the theoretical probability of a randomly selected eastbound driver
getting a green light?
b) Design a simulation you could use to determine the experimental
probability of getting a green light. Carry out the simulation.
Solution
a)
b)
Since the light is programmed to be green 75% of the time,
there is a 75% chance the light will be green for the driver.
The probability is 75%, or 0.75.
One way to simulate this situation is to place 4 slips of paper
into a box.
On 3 of the slips write green.
On the fourth slip write not green.
As in the situation being modelled,
Probability of Green = 34 , or 0.75
Pull out one slip, record the result in a table, and replace the paper
in the box. Repeat the experiment many times.
Yazan completed 20 trials of this simulation.
These are his results.
Event
Tally
Frequency
Green
|||| ||||
|||| ||
17
|||
3
Not green
Experimental probability of Green:
17
= 0.85, or 85%
20
5. Kieran performed the simulation experiment
from the Guided Example and got these results.
a) What are the frequencies?
How many trials is this all together?
b) Determine the experimental probability
of getting a green light.
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Event
Tally
Green
||||
||||
||||
Not green
|||
Frequency
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How does the experimental probability compare to the theoretical probability
in the Guided Example? Do you think Kieran did a better job
conducting the experiment than Yazan? Explain your thinking.
6. A baby has an equal chance of being a boy or a girl.
Think about families you know with 3 children.
Do many of them have all girls or all boys? Is this what you would expect?
b) Draw a tree diagram to show all the possible combinations of girls and boys
for a family with three children.
What is the probability all three children are the same sex?
c) You can simulate this situation with a coin toss.
Let heads represent a girl and tails a boy. On each trial, toss 3 coins.
Complete at least 20 trials. Record your results in a table like this.
a)
Event
Tally
You will need
3 coins.
Frequency
3 girls or 3 boys
Another combination
What was the experimental probability of three girls or three boys?
How does this compare to the theoretical probability?
e) Combine your results with those of several classmates.
Draw a graph to show how the experimental probability changes
as the number of trials increases. Describe the shape of the graph.
d)
7. For each situation, determine the theoretical probability.
Then describe an experiment you could conduct to
simulate each situation.
Explain how you know the simulation correctly models
the situation.
How many trials would you conduct in your simulation?
Explain your reasoning.
a) You have no idea what the answer is to
a multiple-choice question with four answers.
What is the probability you will guess correctly?
b) A scratch and win card has six dots.
Only one dot has WIN under it.
You can scratch one dot.
If the word WIN appears, you win a prize.
What is the probability of choosing the correct dot?
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8. a) Use this frequency distribution graph.
What were the 5 most commonly used letters in the sample of text, in order?
b) The most commonly used letters in three languages are:
Italian: a, e, i, o, l, n, r
Frequency of Letters in a Sample of Text
100
German: e, n, i, r, s, a, h
93
Swedish: e, a, n, t, r, s, i
80
In which of these languages
65
60
do you think the sample
of text was written?
40
36
35
29
27
27
Describe how you
23
23
20
19
18
17
20
13
decided which language
6
8
8
2
1
1 1
1
was the most likely.
0
Number of occurrences
Chapter 08
10
0 0 3
a b c d e f g h i j k l m n o p q r s t u v w x y z
Letter
9. Take it Further One golden ping pong ball is placed in a box
filled with white ping pong balls.
Without looking, contestants reach into the box and pull out one ball.
If the contestant pulls out the golden ball, he or she wins a prize.
After each attempt, the ball is replaced in the box.
a) What additional piece of information would you need to calculate
the theoretical probability of winning a prize?
b) Last week, 539 people tried this contest.
Ninety-six lucky contestants drew the golden ball.
What is the experimental probability of winning a prize?
State your answer as a fraction, percent, and a decimal.
c) About how many white balls do you think are in the box?
Explain the strategy you used to solve this problem.
Justin plans to conduct 50 trials of a coin toss experiment.
After 12 rolls, Justin notices he has tossed 6 heads and 6 tails.
Since the experimental probability of tossing a tail is equal
to the theoretical probability, he decides to stop.
Explain whether Justin was right to stop the experiment early.
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Comparing Theoretical and
Experimental Probability
Graphing calculators and computer spreadsheets have random number generators
that you can use to simulate probability experiments.
This allows you to conduct hundreds of trials in a short time.
Inquire
Simulating Coin Tosses using a Spreadsheet
or Graphing Calculator
The theoretical probability of tossing a head with a coin is 12 .
When you determine the experimental probability, it may be quite different from 0.5.
You will use technology to simulate 10, 100, and 1000 coin tosses.
You will determine the experimental probability of getting heads and graph
this probability as the number of tosses increases.
Choose Using a Spreadsheet or Using a Graphing Calculator.
Using a Spreadsheet
Open the file ProbSim.xls.
To begin, this spreadsheet is set to simulate an experiment
with only 1 outcome.
So, the probability is 1, all the random numbers representing
the outcomes are 1, and the experimental probability is 1.
Cell B1 shows the theoretical probability
of the event being simulated.
Cell C1 shows the number of possible outcomes.
The numbers in cells B6 to B15 represent the
outcomes of 10 trials. The 1s represent successful trials.
The formula in cell B2 counts and displays the number
of 1s in the trials.
The formula in cell B3 calculates the experimental probability by
dividing the number of successes (from cell B2) by the number
of trials (10).
If necessary, click on
the Sheet 1 tab to
display the first
worksheet.
{
If you see error messages in
some cells, go to the Tools
menu and click Add-Ins.
Check the box beside
Analysis ToolPak, then
click OK.
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To change to a coin toss simulation, click on cell B1.
Since the probability of tossing a head is 12 , type the formula =1/2,
then press Enter.
As soon as you press Enter, the numbers
in cells B6 to B15 will change.
Some will probably be 1s and some 2s.
The 1s represent successful trials (heads),
the 2s represent tails.
Your data will likely be different
from those shown in this sample screen.
1. What is the experimental probability of
heads for your 10 trials?
2. Recalculate the spreadsheet by pressing the Calculate Now key, F9.
A new set of random numbers is generated.
What is the new experimental probability?
Click on the Sheet 2 tab to move to the second worksheet.
It simulates 100 trials.
The probability is set to match whatever you entered in cell B1 on Sheet 1.
3. How many of the 100 tosses are heads?
What is the experimental probability of heads for your 100 trials?
4. Recalculate the spreadsheet.
What is the new experimental probability for your 100 trials?
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If you are using a
Macintosh computer and
the F9 key does not
recalculate your file, hold
down the Command key
and press = instead.
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Click on the Sheet 3 tab to move to the third worksheet.
Each of sheets 3, 4, 5, and 6 simulates 250 tosses at a time.
5. How many of the 250 tosses on Sheet 3 are heads?
What is the experimental probability of heads for your 250 trials?
6. Recalculate the spreadsheet.
What is the new experimental probability for the 250 trials?
Click on the Graph tab to move
to the final worksheet.
Rows 2, 3, and 4 show the data from
Sheets 1, 2, and 3, respectively.
Rows 5, 6, and 7 show
combined data for 500, 750,
and 1000 trials.
{
{
The graph shows both
the experimental
and theoretical probabilities.
The data and graph
on your worksheet
will likely be different from those on
the sample shown here.
7. Describe the graphs representing the
theoretical and experimental probabilities.
8. a) When is the experimental probability closest
to the theoretical probability?
When is it farthest away?
b) Recalculate the spreadsheet.
All the experimental probabilities will change, as will the graph.
Does your answer to part a change?
Recalculate a few times. Generally,
when is the experimental probability
If you are able to print, click on the
closest to the theoretical probability?
graph to select it. From the File menu,
select Print. Verify that Selected chart
When is it farthest away?
is chosen, then click OK.
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This spreadsheet was set up to be used for many simulations.
When you simulate a new
situation, you will have to
change the title of the graph.
To simulate rolling a 1 on a die, go back to Sheet 1, click on cell B1,
and change the probability.
Since the probability of rolling a 1 is 16 , type the formula =1/6, then press Enter.
9. Use the spreadsheet to explore how the theoretical and experimental probability
of rolling a 1 compare as the number of trials increases.
Describe your findings.
10. Suppose you wanted to use the spreadsheet
to simulate the pointer landing
on blue on this spinner.
a) What would you enter in cell B1?
Why?
b) What would a 1 represent in this situation?
Using a Graphing Calculator
You will need a TI-83 or TI-84 graphing calculator.
The random integer operation generates and displays a random integer within a range
you specify. You can also specify how many random integers you want.
1. To try out the randInt operation, press d ~ ~ ~ 5.
Make sure your calculator is
The word randInt( appears on the screen.
not set to display numbers
Press 0 ¢ 5 ¢ 7 ¤ Í.
to a particular number of
decimal places. Press
a) What is displayed on the screen?
z † Í.
b) How many numbers were generated?
c) What do you think the range is? How do you know?
d) Suppose you want to get a list of 10 random numbers between 1 and 6.
What do you think you should enter after randInt?
Try your idea.
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Using the Application PROBSIM
PROBSIM is an application that simulates a variety of
probability situations.
It should be preloaded on your graphing calculator.
➢ To begin, you will be simulating tossing a coin.
Prepare a table like the one below.
You will record the data from the calculator simulations in
the table.
Number of trials
Frequency of
heads
Experimental
probability
10
100
250
500
750
1000
Press O.
Use the down arrow key to move
to PROBSIM, then press Í
Í.
A list of the experiments that can
be simulated appears.
To use the commands along the
bottom of the screen, you use the
5 keys directly below the screen.
Press 1 to select Toss Coins.
Then press q to view the
settings screen.
Set the number of trials to 10.
The rest of the settings should
match those shown here.
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Press s to select the OK
command.
Press p to select the
TOSS command.
Ten coin tosses will be
simulated. The frequencies of
tails and heads are graphed.
The graph on your calculator’s
screen will likely be different
than that shown here.
Press ~ ~ to display the
frequency of heads.
Record the frequency in the
table you prepared.
The frequency for your
simulation will likely be
different than that shown here.
<Catch u08_t29>
➢ The data for the first 10 trials is stored.
By performing another 90 trials, you will have data for 100 trials
Press q to return to the Settings screen.
Change the number of trials to 90.
Press s to return the simulation screen, then p to begin the coin tosses.
After the simulation ends, use the arrow keys to check
the frequency of heads.
Record the frequency in the table you prepared.
➢ Press q. Change the number of trials to 150.
Combined with the earlier 100 trials, you will have results for 250 trials.
Press s, then p.
Once the simulation ends, check the frequency of heads.
Record the frequency in the table you prepared.
➢ Press q. Change the number of trials to 250.
Combined with the earlier 250 trials, you will have results for 500 trials.
Press s, then p.
Once the simulation ends, check the frequency of heads.
Record the frequency in the table you prepared.
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➢ Press p to toss another 250 coins.
Check the frequency of heads and record it in your table.
Repeat this step to complete the frequency column of your table.
➢ Press o o s o to end the simulation and the PROBSIM application.
You will lose the results of the trials,
so ensure you have recorded the frequencies.
2. Complete the table by calculating the experimental probabilities.
How do they change as the number of trials increases?
Follow these steps to graph the experimental probabilities from your table.
For comparison, you will graph the theoretical probability of tossing a head, 0.5.
Press … 1 to go to the list
screen.
If L1 is not empty, move to the
L1 heading and press ‘.
Enter the number of trials in
L1. Press Í after each
number to move to the
next row.
Move to L2. Clear the list.
Enter your experimental
probabilities in L2.
Move to L3. Clear the list.
Enter the theoretical
probability 0.5 six times in L3.
The experimental
probabilities for your
simulation will likely
be different than those
shown here.
Press y o to go to the
STAT PLOTS screen.
Press 1 Í † D Í
†y1Í
y2Í
D D Í.
This sets up the graph
of the experimental
probabilities.
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Press y o to return to the
STAT PLOTS screen.
Press 2 Í † D Í
†y1Í
y3Í
D D Í.
This sets up the graph
of the theoretical
probability.
To draw and view the
graphs, press s
q 9.
Describe your graphs representing the theoretical and experimental probabilities.
b) When is the experimental probability closest to the theoretical probability?
When is it farthest away?
3. a)
4. To simulate rolling a die, start the PROBSIM application again.
Use the Roll Dice simulation.
Explore how the theoretical and experimental probability of rolling a 1
compare as the number of trials increases.
Describe your findings.
5. Suppose you wanted to use PROBSIM
to simulate the pointer landing on blue on this spinner.
a) Which simulation would you use?
b) What could represent blue in your simulation?
Reflect
➢ Describe how you could use technology to simulate a probability
question from another section of this chapter.
➢ What are the advantages of using technology to simulate
probability problems?
Are there any disadvantages? If so, what are they?
➢ Which technology would you choose for this section if both were
available? Explain the reason for your choice.
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Designing a Probability Game
Some of the most popular board games have been created by amateur inventors.
An unemployed architect, two Canadian journalists, a waiter, an actress, and three college
friends – all of these people have invented games that sold more than a million copies.
Inquire
Creating a Game
Work with a partner or in a group to design a game.
You can use dice, spinners, counters, cards, coins, and other materials.
1. Planning your game
Your game must involve probability and should be fun to play.
As you design the game, think about these ideas:
➢ Will your game have a topic or theme?
➢ How many players will there be? Will your game be suitable for people
in many age groups or will you target one group?
➢ Will the game be fair or will one player have a greater chance
of winning than the other player or players?
➢ Will the game be entirely based on chance or will skill or strategy also have a role?
If you have trouble thinking of ideas, look back through
probability activities in this chapter. You can also think
about games you have played. After discussing your
ideas, choose one to try out.
➢ Decide on a set of rules for how the game
will be played.
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2. Determining theoretical probability
You might use a computer to
➢ Calculate any theoretical probabilities you can for the game.
display your data.
For example, if each turn involves using a spinner, calculate
the probability of the pointer landing on each section of the spinner.
3. Testing your game
➢ Collect or make the materials you need to try out the game.
Or, use technology to simulate playing the game.
➢ Try out your game or simulation a few times.
Make any changes you feel are necessary to improve the game.
4. Determining experimental probability
➢ Once you are happy with the game, play it a few times.
Keep a record of what happens on each turn.
Use these data to determine experimental probabilities
for events in the game.
➢ If the experimental probabilities are very different from
what you expected, check that you followed the rules of
your game correctly and that your materials are fair.
Check your theoretical calculations.
5. Presenting your game
➢ Present your game to another group.
Describe how to play the game, any surprises
you encountered when you tried the game,
and any revisions you made.
➢ Invite the group members to play your game a few times.
Ask whether they enjoyed it and would play it again.
Invite them to make suggestions for improvements.
➢ Change roles and try out the other group’s game.
Reflect
➢ Describe one way your game uses math.
➢ Describe a decision you made when creating your game.
How did you make the decision?
➢ What probability strategy might someone use in the game?
How might using this strategy affect the game?
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Chapter Review
What Do I Need to Know?
Probability
Probability is a way of describing the likelihood of an event happening.
Probability can be described using words, or numbers between 0 and 1 and
can be expressed as a fraction, decimal, or percent.
Theoretical Probability
Theoretical probability is based on an analysis of the possible outcomes.
It is often just called probability.
When you calculate the theoretical probability of an event, you must:
count the possible outcomes;
know that the outcomes are equally likely; and
count the outcomes that are favourable to the event.
You can then use this formula:
outcomes favourable to the event
Probability of an event = Number of
Total number of outcomes
Experimental Probability
Experimental probability is calculated using data from an experiment or survey.
of times the event happens
Experimental probability of an event = NumberTotal
number of trials
When you conduct many trials of an experiment, the experimental probability tends
to approach the theoretical probability.
Probability Simulations
The probability of real-life situations can be simulated using dice, spinners,
coins, and other models. A probability simulation must have the same probability
as the situation it models.
Spreadsheets and graphing calculators have random number generators that
enable them to simulate many probability situations.
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What Should I Be Able to Do?
8.1
Which two types of restaurants are
you equally likely to pick?
Explain your reasoning in each part.
1. A mother tongue is the first language a
c)
person learns. The mother tongue data in
the table are rounded to the nearest
percent.
3. A recent survey found that 171 out of 225
Mother tongue
(% of population)
Province/
territory
English
French
high school students prefer to play sports
than to watch sports on television. What
is the probability that a student chosen
at random would prefer to play sports
than to watch them on television?
Express your answer as a fraction,
percent, and decimal. How did you
determine your answers?
Other
Alberta
82
2
16
New
Brunswick
65
33
2
Nunavut
27
2
71
For a survey, suppose you want to
contact people outside Québec
whose mother tongue is French.
In which province or territory in the
table would you have the greatest
chance of finding them?
b) One of these places has four official
languages: English, French, Inuktitut,
and Inuinnaqtun.
Which province or territory do you
think it is? Explain your answer
c) Write a question involving probability
that someone could answer using
these data.
a)
8.2
4. The seats in a small plane are arranged as
shown below. Suppose you are the first
passenger to buy a ticket and are assigned
a seat at random.
Determine the probability that your seat
satisfies each condition.
Arrange the conditions from most likely
to least likely. Explain your strategy.
a) It is a window seat.
b) It is in the rear of the plane, behind
the passenger entrance.
c) It is on the port (left) side of the
plane.
d) It is in an emergency exit row.
2. A page in a restaurant guide has ads for
these types of restaurants: 7 Italian,
4 Asian, 2 Mexican, and 4 Indian. All the
ads are the same size and they fill the page.
Suppose you were to pick a restaurant by
closing your eyes and pointing to the page.
a) What type of restaurant are you most
likely to pick?
b) What type of restaurant are you least
likely to pick?
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Emergency exit
Port
Passenger
entrance
Emergency exit
Starboard
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5. A marble is drawn at random from a box
Get a die. Roll it at least 20 times.
What is the experimental probability
of rolling a 6?
d) If the technology is available, use a
spreadsheet or graphing calculator to
simulate rolling a die 250 times. What
is the experimental probability of
rolling a 6? Graph the probability for
different numbers of trials.
c)
containing 21 red, 17 green, 14 yellow,
and 20 blue marbles. What is the
probability of each event? Express your
answers as decimals.
a) Picking a blue marble
b) Picking a purple marble
c) Picking a green or yellow marble
8.2
8.3
8.3
6. Prepare a concept map or a set of strong
notes about probability.
7. a) List all the ways the coins can land if
you toss a penny, a nickel, and a dime
at the same time. Describe what tool
you used to organize your list.
b) What is the probability of tossing
exactly 2 tails? How do you know?
c) Get a penny, a nickel, and a dime and
toss them at least 20 times. Keep track
of the number of heads and tails
tossed each time.
What is the experimental probability
of tossing exactly 2 tails?
d) Suppose you could collect data for 200
tosses. How might the experimental
probability change? Explain your
thinking.
8.3
8.4
8.4
9. The graph shows data from a spreadsheet
simulation of a probability experiment.
a) What was the theoretical probability?
b) What experiment might the
spreadsheet be simulating?
Explain your thinking.
c) Describe how the experimental
probability changed as the number of
trials increased.
8. Catherine is playing a board game that
uses a die. On her next turn, she will win
if she rolls a 6.
a) What is the probability that Catherine
will win the game on her next roll?
b) Would your answer to part a change if
you knew Catherine had rolled a 6 on
each of her last 2 turns?
Explain why or why not.
8.5 10. A weighted die is designed so that one
number is more likely to be rolled than
the others. You purchase a fair die and a
weighted die from a joke shop.
Unfortunately, you lose track of which die
is which. Describe how you could find
out.
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Practice Test
Multiple Choice: Choose the correct answers for questions 1 and 2. Justify each choice.
1. You toss 2 coins. Which of these outcomes is most likely?
Two heads
C. One head and one tail
A.
Two tails
D. All outcomes are equally likely
B.
2. A company is handing out free sample bottles of 6 flavours of juice.
How could you simulate receiving your favourite flavour, cranberry?
A. Roll a die many times, letting 1 represent receiving a bottle of cranberry juice.
B. Put 6 pieces of paper, 1 marked “cranberry,” in a cup.
Repeatedly draw and replace a slip without looking.
C. Either of these simulations would be suitable.
D. Neither of these simulations would be suitable.
Show your work for questions 3 to 6.
3. Thinking A student says: “Theoretical probability tells us that if you toss a coin 100 times,
you will get 50 heads and 50 tails.” Is the students’ reasoning correct?
Explain why or why not.
4. Knowledge and Understanding A bag contains 3 green marbles, 2 yellow marbles,
and 1 white marble.
a) What is the theoretical probability of picking a yellow marble?
b) Draw a spinner that could be used to simulate this situation.
Use the spinner to determine the experimental probability.
How does it compare to the theoretical probability?
Do you still think your spinner is correct? Explain.
You will need a paper
clip to use as a
pointer.
5. Application Suppose you are a hockey coach. Your team has 2 goalies.
Goalie A stopped 81 of the 93 shots she has faced this season.
Goalie B stopped 75 of the 82 shots she has faced this season.
Which player would you start in goal for your next game? Explain your choice.
6. Communication A pre-election poll reveals that 48% of voters would vote for
Candidate A, 35% would vote for Candidate B, and 17% would vote for Candidate C.
a) Based on this poll, which candidate is most likely to win the election?
b) Suppose most of Candidate C’s supporters switch their support to Candidate B.
How might this affect the outcome of the election? Explain your reasoning.
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