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PROBABILITY
1. Exploring Probability – pg. 140-141
Assignment: pg. 141 #1-4
2. Probability and Odds – pg. 142-150
Assignment: pg. 148-150 #1-16
3. Probabilities Using Counting Methods – pg. 151-162
Assignment: pg. 159-162 #1-15 (omit 9 and 14), 18
4. Mid-Chapter Review – pg. 163-165
Assignment: pg. 165 #1-9
5. Mutually Exclusive Events – pg. 166-180
Assignment: pg. 176-180 #1-9, 13, 17, 18
6. Conditional Probability – pg. 182-191
Assignment: pg. 188-191 #1-9, 13-15, 18
7. Independent Events – pg. 192-201
Assignment: pg. 198-201 #1-13, 16
8. Chapter Quiz
9. Chapter Review – pg. 203-206
Assignment: pg. 205-206 #1-19 (omit 14, 15)
10. Chapter Exam
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EXPLORING PROBABILITY
Learning Outcomes:

To understand the difference between experimental and theoretical probability

To use probability to make predictions
Recall:
𝑃(𝑒𝑣𝑒𝑛𝑡) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
Probability is expressed as a value between 0 and 1, not a percent.
The set of all possible outcomes of a particular experiment is called the sample space.
An event is any collection of possible outcomes of an experiment, that is, any subset of the
sample space.
A trial is any operation whose outcome cannot be predicted with certainty.
An outcome is the result of carrying out an experiment.
An impossible event will yield a probability of 0 and a certain event will yield a probability of
1.
Complete the following investigation with a partner:
You will need 2 dice and some graphing paper.
You and your partner are playing a game. You roll two dice and determine the difference
between the numbers rolled.
Before you attempt any rolling of the dice, complete the following questions:
1. Predict the following:

Which difference will occur the most often?

Which difference will occur the least often?

Which differences will occur equally often?
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2. Complete the experiment. Roll two dice 36 times. Record the difference each time as a
positive value. Record your values below. How closely do your results match your predictions?
3. Represent your results using a bar graph.
4. Determine the experimental probability for each difference as a fraction.
Times occurred out of 36
3
5. Compare your results with another group’s experimental results. Were they similar?
Combine the two sets of results, and repeat Part 4 for the combined results. How do these
probabilities compare with those from part 4?
6. Copy and complete this outcome table. Record the differences as positive values.
Differences in Rolls of Two Dice
Die1/Die 2
1
1
0
2
3
4
5
6
2
3
4
5
6
7. Determine the theoretical probability of each difference as a fraction. How closely does the
experimental probability match the theoretical probability?
𝑃(0) =
,
𝑃(1) =
𝑃(4) =
,
,
𝑃(2) =
𝑃(5) =
4
, 𝑃(3) =
,
𝑃(6) =
,
Investigation #2
With a partner, draw a tree diagram to show all the possible outcomes of rolling two regular sixsided dice:
Outcomes for Two Dice
1. What is the total number of possible outcomes when rolling two six-sided dice?
2. How many outcomes have double numbers?
3. What is the probability of rolling two different numbers?
Complementary event: of 6
4. Which difference is impossible?
The event “rolling two different numbers” is called the complement of “rolling double numbers.”
These two events are called complementary events. The probability of an event and the
probability of the complement will always add up to 1.
Events formed from repeated trials or from a combination of simple events are called compound
events and often a table, chart, or tree diagram is useful in determining the sample space.
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Ex. a. List the sample space of an experiment in which one coin is tossed and one six-sided die
is rolled at the same time.
Create a tree diagram
H
T
b. Determine the probability of the event of tossing a head and rolling a 5.
c. Determine the probability of the event of tossing a head and rolling either a 4 or a 2.
The probability of an event can range from 0 (impossible) to 1 (certain). You can express
probability as a fraction, a decimal, or a percent.
Assignment: pg. 141 #1-4
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PROBABILITY AND ODDS
Learning Outcomes:

To explain, using examples, the relationship between odd (part-part) and
probability (part-whole)

To be able to express odds as probability and vice versa

To determine the probability of, or the odds for and against, an outcome in a
situation

To explain, using examples, how decisions may be based on probability or odds
and on subjective judgments

To solve a contextual problems that involves odds or probabilities
You can calculate the chance, or probability that a particular event will happen by finding the
ratio of the number of ways the event can occur to the number of possible outcomes. The
probability of an event may be written as a fraction, decimal, or percent. When outcomes have an
equal chance of occurring, they are equally likely. When an outcome is chosen without any
preference, the outcome occurs at random.
The odds of an event occurring is the ratio of the number of ways the event can occur
(successes) to the number of ways the event cannot occur (failures).
Odds = successes : failures
As an example, out of 25 people, 10 are teens. The odds that a person is a teen would be 10
(successes) to 15 (failures).
10 : 15
Odds in favour: The ratio of the probability that an event will occur to the probability that the
event will not occur, or the ratio of the number of favourable outcomes to the number of
unfavourable outcomes.
Odds against: The ratio of unfavourable outcomes to the number of favourable outcomes.
Definition of
Probability
Definition of
Odds
𝑃(𝑒𝑣𝑒𝑛𝑡) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Odds of an event = number of ways the event can occur : number of ways the event cannot occur
= successes : failures
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Ex. Find the odds of randomly selecting the letter p in the word “Mississippi.”
11 total letters, 2 are p’s.
Ex. What are the odds of randomly selecting a dime from a dish containing 11 pennies, 6
nickels, 5 dimes, and 3 quarters?
Total coins: 25, not dimes 25 – 5 = 20
`
Ex. a. Eleven poker chips are numbered consecutively 1 through 10, with two of them labeled
with a 6 and placed in a jar. A chip is drawn at random. Find the probability of drawing a 6.
b. What are the odds in favour of the event happening?
c. What are the odds against the event happening?
Ex. Suppose that the odds in favour of an event are 5:3. Is the probability that the event will
happen greater or less than 50%?
Total outcomes: 5 + 3 = 8
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Ex. A hockey game has ended in a tie after a 5 min overtime period, so the winner will be
decided by a shootout. The coach must decide whether Ellen or Brittany should go first in the
shootout. The coach would prefer to use her best scorer first, so she will base her decision on the
player’s shootout records.
Player
Ellen
Brittany
Attempts
13
17
Goals Scored
8
10
Who should go first?
Find the percentage of success of each player:
𝐸𝑙𝑙𝑒𝑛:
𝐵𝑟𝑖𝑡𝑡𝑎𝑛𝑦:
Ex. A group of grade 12 students are holding a charity carnival to support a local food bank.
The students have created a dice game that they call Bim and a card game that they call Zap.
The odds against winning Bim are 5:2, and the odds against winning Zap are 7:3. Which game
should Nolan play?
P(A’) is the probability of the complement of A, where P(A’) = 1 – P(A)
If the probability of winning is 70%, the complementary event (opposite) would be the
probability of losing, which is 100 – 70 = 30%
Odds: Part : Part
Probability: Part – whole
Assignment: pg. 148-150 #1-16
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PROBABILITIES USING COUNTING METHODS
Learning Outcomes:
 Be able to solve a contextual problem that involves probability and permutations
 Be able to solve a contextual problem that involves combinations and probability
Review:
Factorial Notation: 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2)(𝑛 − 3) ….
Permutations:
n!
P

n r (n  r )!
Permutations with Restrictions:
n!
Combinations: n Cr 
 n  r !r !
n!
a !b !c !
Explain the difference between combinations and permutations:
Permutations: order important
Ex. Two cards are picked without replacement from a deck of 52 playing cards. Determine the
probability that both are kings.
Is order important?
P( K1 and K 2 ) = favourable outcomes
total outcomes
Ex. The athletic council decides to form a sub-committee of seven council members to look at
how funds raised should be spent on sports activities at the school. There are a total of 15
athletic council members, 9 males and 6 females. What is the probability that the sub-committee
will consists of exactly 3 females?
P(3F) = # of ways to choose exactly three females
# of ways to choose the sub-committee
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Ex. A bag of marbles contains 5 red, 3 green, and 6 blue marbles. If a child grabs three marbles
from the bag, determine the probability that:
a. exactly 2 are blue:
P(2B)= P(2B and 1 other)
= # of ways to choose 2 blue and 1 other
# of ways to choose any three marbles
a. at least one is blue:
P(1B and 2 other) or P(2B and 1 other) or P(3B and 0 others)
b. the first is red, the second is green and the third is blue:
is order important?
P(1R and 1G and 1B)
c. One is red, one is green and one is blue:
is order important?
P(1R and 1G and 1B)
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Ex. To win a prize at a local radio station, a contestant needs to spell out the word
SASKATOON with letter tiles. If the tiles are mixed up and all face down, what is the
probability that the contestant will win the prize?
Only one way to spell Saskatoon (1 favourable outcome)
Total outcomes is:
The probability of winning is
Ex. The letters of the word OKOTOKS are arranged. The probability, to the nearest hundredth,
that an arrangement, chosen at random from all possible arrangements, begins and ends with a
vowel is:
Assignment: pg. 159-162 #1-15 (omit 9, 14), 18
Mid-Chapter Review – pg. 163-165
Assignment: pg. 165 #1-9
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MUTUALLY EXCLUSIVE EVENTS
Learning Outcomes:

To classify events as mutually exclusive or non-mutually exclusive, and explain
the reasoning

To determine if two events are complementary, and explain the reasoning

To represent mutually exclusive events and non-mutually exclusive events

To solve a contextual problem that involves the probability of mutually exclusive
or non-mutually exclusive events



To solve a problem that involves the probability of complementary events
Events are said to be mutually exclusive if they have no common outcomes.
For mutually exclusive events we are looking for the word “OR”
Recall Venn Diagrams:
A mutually exclusive event states that there are no common outcomes. How could a mutually
exclusive event be represented using a Venn Diagram?
How could a non-mutually exclusive event be represented using a Venn Diagram?
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Investigation
Consider the experiment of drawing a card from a regular a 52 card deck. Let the event A be “a
heart is drawn” and event B be “a spade is drawn.”
1. Mark the outcomes to the experiment on the Venn Diagram which
represents the sample space. Because these events have _____
common outcomes, we say the events A and B are _________
_______.
Use the following formulas for the probability of the event 𝑨 ∪ 𝑩
If the events A, B are mutually exclusive, then: 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
If the events A, B are NOT mutually exclusive, then: 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
2. Determine the P(A ∪ B)
P (A or B) = P(A) + P(B)
= 0.25 + 0.25 = 0.5
Basic strategies when dealing with mutually exclusive events:

If mutually exclusive, add the individual probabilities

If not mutually exclusive, a venn diagram is useful for organizing our information
If you add the percentages when solving a problem and you find your solution is over 100% or 1,
this will tell you that you are not dealing with a mutually exclusive event.
Examples
1. State whether the events A and B are mutually exclusive or not.
a) Experiment – a card is drawn from a standard deck
Event A – a face card is selected
Event B – a diamond is selected
Ask: Do the two events have anything in __________________?
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b) Experiment – two dice are thrown
Event A – the dice both show the same value
Event B – the total score is 8
c) Experiment – two dice are thrown
Event A – the dice both show the same value
Event B – the total score is 9
2. A single die is rolled. What is the probability of rolling a 2 or a 6?
P(2) + P(6) =
3. A single card is drawn from a standard deck of cards. What is the probability of drawing
a red card or a black queen?
4. A traffic light on the side of a busy street is green for 25 seconds, yellow for 5 seconds
and red for 30 seconds for every minute during the rush hour traffic from 4:00 p.m. to
6:00 p.m. What is the probability that the traffic light will not be green when a motorist
first sees it during rush hour traffic?
Solution 1: Using mutually exclusive events
P( yellow or red) =
Solution 2: Using complementary probability
𝑃(𝑔𝑟𝑒𝑒𝑛
̅̅̅̅̅̅̅̅) = 1 − 𝑃(𝑔𝑟𝑒𝑒𝑛) =
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Ex. A school newspaper published the results of a recent survey of students’ eating habits. 62%
said they skip breakfast, 24% skip lunch, and 22% eat both breakfast and lunch.
a. Are skipping breakfast and lunch mutually exclusive events?
b. Determine the probability that a randomly selected student skips breakfast but not
lunch. (You may find using a venn diagram helpful)
B
L
54
8
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c. Determine the probability that a randomly selected student skips at least one of
breakfast or lunch.
Skips at least one of breakfast or lunch means the same as skips one or the other or both.
Ex. Max’s mother buys a new washer and dryer set for $2500 with a 1-year warranty. She can
buy a 3-year extended warranty for $450. Max researches the repair statistics for this washer and
dryer set and finds the data in the table below. Should Max’s mother buy the extended warranty?
Appliance
P( repair within extended
Average
warranty period)
Repair Cost
Washer
22%
$400
Dryer
13%
$300
Both
3%
$700
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Ex. A car manufacturer keeps a database of all the cars that are available for sale to all the
dealerships in Western Canada. For model A, the database reports that 43% have heated seats,
36% have a sunroof, and 49% have neither. Determine the probability of a model A car at a
dealership having both heated seats and a sunroof.
Assignment: pg. 176-180 #1-9, 13, 17, 18
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CONDITIONAL PROBABILITY
Learning Outcomes:

To determine the probability of an event, given the occurrence of a previous event

To explain, using examples, how decisions may be based on probability on
subjective judgments

To solve a contextual problem that involves probability
Investigate
A computer manufacturer knows that, in a box of 100 chips, 3 will be defective. Jocelyn will
draw 2 chips, at random, from a box of 100 chips.
What is the probability that both the chips will be defective?
1. Draw a Tree Diagram to represent the ways you can draw two computer chips from the box.
2. Name the four permutations for the situation.
3. Which of the 4 permutations are we concerned with?
4. What is the probability of drawing a defective chip on the first pull?
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5. What is the probability of drawing a defective chip on the second pull? Does the first draw
impact the probability of the second pull? If so, how?
6. How can I determine the probability of drawing two defective chips? What is that
probability?
7. How does the probability that the first event will occur affect the probability that the second
even will occur? Explain.
8. Suppose that Jocelyn replaced the first chip before drawing the second chip. Would the
probability of the second chip being defective remain the same?
9. When determining the probability of drawing two defective chips, Jocelyn did not consider
the probability that the first chip she drew would not be defective. Explain why.
10. Explain why the probability of drawing a defective chip on the second draw is considered a
conditional probability.
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11. Go back to the tree diagram drawn on step 1. Label each branch with its probability.
Determine the probability of drawing each permutation of defective and not defective chips, then
add these probabilities. What does the sum imply?
Def/Def:
Def/NDef:
NDef/Def:
NDef/NDef:
Dependent Events: Events whose outcomes are affected by each other.
Conditional Probability: The probability of an event occurring given that another event has
already occurred.
If event B depends on event A occurring, then the conditional probability that event B will
occur, given that event A, has occurred, can be represented as follows:
𝑃(𝐵\𝐴) =
𝑃(𝐴 ∩ 𝐵)
𝑃(𝐴)
If event B depends on event A occurring, then the probability that both events will occur can be
represented as follows:
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵\𝐴)
20
Ex. According to a survey, 91% of Canadians own a cellphone. Of these people, 42% have a
smartphone. Determine, to the nearest percent, the probability that any Canadian you met during
the month in which the survey was conducted would have a smartphone.
Let C = owning a cellphone
Let S = owning a smartphone
P(C) = 0.91
P(S\C) 0.42 (conditional on owning a cellphone)
0.42 =
𝑃(𝐴 ∩ 𝐵)
0.91
Ex. A test for Type 2 diabetes (non-insulin dependent) measures the blood glucose level after
eight hours of fasting. Consider a blood glucose level above normal to be a positive result and
anything else to be a negative result. This test is 85% accurate, and 2% of the world’s population
actually has diabetes. Determine:
a. A tree diagram for the situation
0.85
0.02
0.15
0.15
0.98
0.85
b. the probability that an individual tests positive for diabetes.
There are two P(diabetic and test positive):
P(not diabetic and tests positive):
Total:
21
c. the probability of P(diabetic\tests positive)
𝑃(𝑑𝑖𝑎𝑏𝑒𝑡𝑖𝑐\𝑡𝑒𝑠𝑡𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒) =
𝑃(𝑑𝑖𝑎𝑏𝑒𝑡𝑖𝑐 𝑎𝑛𝑑 𝑡𝑒𝑠𝑡𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒)
𝑃(𝑡𝑒𝑠𝑡𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒)
Assignment: pg. 188-191 #1-9, 13-15, 18
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INDEPENDENT EVENTS
Learning Outcomes:

To compare, using examples, dependent and independent events

To determine the probability of two independent events
Investigate: Consider the following situations.
Situation 1
Drawing two cards from a deck
without replacement
A: The first card is a spade
B: The second is a spade
Situation 2
Drawing two cards from a deck
with replacement
A: The first card is a spade
B: The second is a spade
How many cards were in the deck on the first
draw?
How many cards were in the deck on the first
draw?
How many of these cards were spades?
How many of these cards were spades?
How many cards were in the deck on the
second draw?
How many cards were in the deck on the
second draw?
How many of these cards were spades?
How many of these cards were spades?
Calculate P(A and B)
Calculate P(A and B)
These are dependent events
These are independent events
With a partner, explain the differences between a dependent and independent event.
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Dependent Events: Two events are dependent if after the first event has occurred, it effects the
probability of the other event occurring. The probability of event B depends on whether or not
event A occurred.
Independent Events: Two events are independent if after the first event has occurred it has no
effect on the probability of the second event occurring. The probability of event B does not
depend on whether or not event A occurred.
To determine the probability of two events occurring we can use the formula:
P(A and B) = P(A) x P(B)
NOTE: If the events are dependent, you must account for this in the probability of event B
Ex. The probability that Ashley will pass Math this semester is 0.7 and the probability that she
will pass English this semester is 0.9. If these events are independent, determine the following to
the nearest hundredth:
a. Ashley will pass math and English
b. Ashley will pass math but not English
c. Ashley will pass English but not Math
d. Ashley will pass neither
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Ex. Wayne and Kelly are playing a die and coin game. Each turn consists of rolling a regular
die and tossing a coin. Points are awarded for rolling a 6 on the die and/or tossing heads with the
coin:

1 point for either outcome

3 points for both outcomes

0 points for neither outcome
Players alternate turns. The first player who gets 10 points wins.
Determine the probability that Wayne will get 1, 3, or 0 points on his first turn, then
verify your results.
a. Are the events independent?
b. What is the probability of rolling a six and tossing a heads?
Probability of rolling a six:
Probability of tossing a heads:
c. List all the permutations to represent the situation, then determine the probability of each
situation happening.
Outcomes:
Rolling a six and a heads (3 points):
Rolling a six and a tail(one point):
Not rolling a six and head (one point):
Not rolling a six and a tail (0 points):
d. What is the total probability of getting 1, 3 or 0 points on the first turn?
Probability of 1 point:
Probability of 3 points:
Probability of 0 points:
Total:
25
Ex. All 1000 tickets for a charity raffle have been sold and placed in a drum. There will be two
draws. The first draw will be for the grand prize, and the second draw will be for the consolation
prize. After each draw, the winning ticket will be returned to the drum so that it might be drawn
again. Max has bought five tickets. Determine the probability, to the nearest tenth of a percent,
that he will win at least one prize.
3 situations: win both, grand prize, consolation prize
Probability of winning grand prize or consolation prize:
Outcomes possible: win both, win grand prize only, or win consolation prize only
Probability of winning both:
Probability of winning grand prize only:
Probability of winning the consolation prize only will be the same as winning grand prize only.
Total probability:
Assignment: pg. 198-201 #1-13, 16
Chapter Quiz
Chapter Review – pg. 203-206
Assignment: pg. 205-206 #1-19
Chapter Exam
26