Chapter 3: Probability

Chapter 3: Probability Section 3.1

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Chapter 3: Probability

Section 3.1: Exploring Probability

Terminology:

Experimental Probability:

The probability that an outcome will occur when measured from a set of

experimental results.

The experimental probability of an event A is represented as:

𝑃(𝐴) =𝑛(𝐴)

𝑛(𝑇)

where n(A) is the number of times that event A occurred and n(T) is the total

number of trials, T, in the experiment.

Theoretical Probability:

The probability that an outcome will occur when measured from the total number

of possible outcomes.

The theoretical probability of an event A is represented as:

𝑃(𝐴) =𝑛(𝐴)

𝑛(𝑆)

where n(A) is the number of times that event A occurred and n(S) is the total

number of outcomes in the sample space, S, where all outcomes are equally likely.

Fair Game:

A game in which all players are equally likely to win; for example, tossing a coin

to gets heads or tails is a fair game.

NOTE:

- An event is a collection of outcomes that satisfy a specific condition. For example,

when throwing a standard die, the event “thrown an odd number” is a collection

of outcomes 1,3, and 5.

- The probability of an event can range from 0 (impossible) to 1 (certain). You can

express probability as a fraction, decimal, or percentage.

- You can use theoretical probability to determine the likelihood that an event will occur.


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Section 3.2: Probability and Odds

Terminology:

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.

𝑛(𝐴): 𝑛(𝐴′)

Odds Against:

The ratio of the probability that an event will not occur to the probability that the

event will occur, or the ratio of the number of unfavourable outcomes to the

number of favourable outcomes.

𝑛(𝐴′): 𝑛(𝐴)

Determining Odds Using Sets

Ex1. Baily holds all the hearts from a standard deck of 52 playing cards. He asks Morgan

to choose a single card without looking.

(a) Determine the odds in favour of Morgan choosing a face card.

(b) Determine the odds against Morgan drawing a face card.

(c) Compare the odds of both outcomes, which is more likely to occur?


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Ex2. Matt is playing dice. He is rolling one dice and will win the game if his roll is

greater than 4.

(a) Determine the odds in favour of winning the game.

(b) Determine the odds against winning the game.

Determining Odds from Probability

Ex1. Research shows that the probability of an expectant mother, selected at random,

having twins is 1

32.

(a) What is the odds in favour of the mother having twins?

(b) What are the odds against an expectant mother having twins?


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Ex2. Suppose the probability of having twins was in fact 2

5.

(a) What are the odds in favour of the event happening?

(b) What are the odds against the event happening?

Determining the Probability From Odds

Ex1. A computer randomly selects a university student’s name from the university

database to award a $100 gift certificate from the bookstore. The odds against the

selected student being male are 57:43. Determine the probability that the randomly

selected university student will be male.

Ex2. Suppose the odds in favor of the student being female were 5:3. Determine the

probability of the chosen student being female. Is the probability greater than 50%?


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Making a Decision Based on Odds and Probability

Ex1. 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 players’ shootout records.

Ex2. What is Jose is also on the team and she has scored twice in the last three shootout

attempts. Should the coach let her shoot first, second, or third? Explain.

Player Attempts Goals Scored

Ellen 13 8

Brittany 17 10


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Interpreting Odds Against and Making a Decision

Ex1. A group of Grade 12 students are holding a charity carnival to support a local

animal shelter. 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 Madison play for the best chance to win?

Ex2: The grade 12 student want to include one more game at their charity carnival. They

need to choose between game A and game B. The odds against winning game A are 11:3,

and the odds against winning game B are 17:6. The goal is to raise as much money as

possible for the animal shelter. Which game should the students choose? Assume that

people are equally likely to play the two games.


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SUMMERY:

The probability of an event A not occurring P(A’) is given as 𝑃(𝐴′) = 1 − 𝑃(𝐴)

(where A’ is the complement of A)

If the odds in favour of event A occurring are m:n, then the odds against event A

occurring are n:m.

If the odds in favour of event A occurring are m:n, then 𝑃(𝐴) =𝑚

𝑚+𝑛

Practice Questions

#1-18 pg 148-49


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Section 3.3: Probabilities Using Counting Methods

Solving a Probability Problem Using Counting Techniques

Ex1. Jamaal, Ethan, and Alberto are competing with seven other boys to be on their

school’s cross-country running team. All the boys have an equal chance of winning the

trial race. Determine the probability that Jamaal, Ethan, and Alberto will place first,

second, and third, in any order.

Ex2. Suppose Zachary is also trying out for the team, so now there will be 11 runners in

the trial race. What is the probability that three of Jamal, Ethan, Alberto, and Zachary

will place in the top three positions in any order?


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Solving a Probability Problem with the Fundamental Counting Principle

Ex1. About 20 years after they graduated from high school, Nick, Ryan, and Jayden met

in a mall. Nick had two daughters with him, and he said that he had three other children

at home. Determine the probability that at least one of Nick’s children is a boy.

Ex2. Suppose that instead, Jayden had brought one daughter to the mall with him and

said he had four others at home. Determine the probability that:

(a) All five of his children are girls.

(b) At least one of his children are boys.


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Solving a Probability Problem Using Reasoning

Ex1. Beau hosts a morning radio show in Saskatoon. To advertise his show, he is holding

a contest at a local mall. He spells out SASKATCHEWAN with letter tiles. Then he turns

the tiles face down and mixes them up. If the row of tiles spells out SASKATCHEWAN,

Sally will win a new car. Determine the probability that Sally will win the car.

Ex2. Suppose that the contest word was changed. Determine the probability of winning

if the word was changed to:

(a) SASKATOON

(b) CANADA


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Solving a Probability Problem with Conditions

Ex1. There are 18 bikes in Marnie’s spinning class. The bikes are arranged in 3 rows,

with 6 bikes in each row. Megan, Morgan, Corey, Erica, Mackenzie, and Kendra each call

the gym to reserve a bike. They hope to be in the same row, but they cannot request a

specific bike. Determine the probability that all 6 friends will be in the same row, with

Erica and Morgan at either end.

What if the gym were rearranged such that there were four rows of five bikes instead

and Meagan decides not to go to class. What is the probability that the 5 friends be in

the same row, with Erica and Morgan at either end.

Practice Questions

#4-17 pg 160-61


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Section 3.4: Mutually Exclusive Events

Terminology:

Mutually Exclusive:

Two events are considered to be mutually exclusive if one does not impact the other. In

other words changing the outcome for the first event has no impact on the outcomes of

the second event. Data for mutually exclusive events can be represented by two disjoint

sets of data in a Venn diagram.

Probability for Mutually Exclusive Events

The probability for mutually exclusive events can be represented using the following:

𝑃(𝐴 ∪ 𝐵) =𝑛(𝐴∪𝐵)

𝑛(𝑈)

Since we know that for two mutually exclusive events, the number of elements in the

union of the two sets is just the sum of the number of elements in A and the number of

elements in B, we can say:

𝑃(𝐴 ∪ 𝐵) =𝑛(𝐴) + 𝑛(𝐵)

𝑛(𝑈)

𝑃(𝐴 ∪ 𝐵) =𝑛(𝐴)

𝑛(𝑈)+

𝑛(𝐵)

𝑛(𝑈)

𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

UBA

The Probability of Mutually Exclusive Events

Given two mutually exclusive events, A and B, the probability of both events happening can be

calculated by determining the sum of the probabilities of each event.

𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)


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Probability for Non-Mutually Exclusive Events

The probability for non-mutually exclusive events can be represented using the

following:

𝑃(𝐴 ∪ 𝐵) =𝑛(𝐴 ∪ 𝐵)

𝑛(𝑈)

Since we know that for two non-mutually exclusive events, the number of elements in

the union of the two sets is just the sum of the number of elements in A and the number

of elements in B subtract the intersect of the two sets, we can say:

𝑃(𝐴 ∪ 𝐵) =𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)

𝑛(𝑈)

𝑃(𝐴 ∪ 𝐵) =𝑛(𝐴)

𝑛(𝑈)+

𝑛(𝐵)

𝑛(𝑈)−

𝑛(𝐴 ∩ 𝐵)

𝑛(𝑈)

𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)

UBA

The Probability of Non-Mutually Exclusive Events

Given two mutually exclusive events, A and B, the probability of both events happening can be

calculated by determining the sum of the probabilities of each event subtracting the probability of

both events occurring.

𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)


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Determining the Probability of Events that are Not Mutually Exclusive

Ex1. Janet and Violet were playing a board game. According to one of the rules, if a

player rolls a sum that is greater than 8 or a multiple of 5, the player gets a bonus of 100

points. Determine the probability that Violet will receive a bonus of 100 points on her

next roll. Assume that she is rolling two dice.


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Using a Venn Diagram to Solve a Probability Problem that Involves Two Events

Ex1. A school newspaper published the results of a recent survey. The survey showed

that 62% of students skip breakfast, 24% of students skip lunch, and 22% eat both

breakfast and lunch.

(a) Are skipping breakfast and skipping lunch mutually exclusive events?

(b) Determine the probability that a randomly selected student skips breakfast but

not lunch.

(c) Determine the probability that a randomly selected student skips at least one of

breakfast or lunch.


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Ex2. Suppose that the school cafeteria introduces a different lunch menu. Now only 14%

of the students skip lunch, but 62% of the students continue to skip breakfast and 38%

eat both breakfast and lunch. Determine the maximum percentage of students that now

skip both breakfast and lunch.

Ex3. Reid’s mother buys a new washer and dryer set for $2500 with 1-year warranty.

She can buy a 3-year extended warranty for $450. Reid researches the repair statistics

for this washer and dryer set and finds the data in the table below. Should Reid’s mother

buy the extended warranty?

Appliance P(repair within

extended warranty period)

Average Repair Cost

Washer 22% $400 Dryer 13% $300 Both 3% $700


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Ex4. Suppose that the average repair she wished to buy a Stove and Refrigerator set for

$3000 with a possible extended 3-year warranty that would cost $500. Should Reid’s

Mother buy the extended warranty for this appliance set?

Appliance P(repair within

extended warranty period)

Average Repair Cost

Fridge 27% $675 Stove 16% $490 Both 10% $1165


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Describing the Probability of Two Events

Ex1. A car manufacturer keeps a database of all the cars that are available for sale at all

dealerships in Western Canada. For model A, the database reports that 43% have heated

leather 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.

Ex2. The database also reports that 56% of model B cars have heated leather seats, 49%

have a sunroof, and 27% have neither. What is the probability of a model B car at a

dealership having a sunroof but no heated leather seats?

Practice Questions

5,6,7,8,9,12,13,14,15,16 pg 177-179


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Section 3.5: Conditional Probability

Terminology:

Dependent Events:

Events whose outcomes are affected by each other.

Ex. If two cards are drawn from a deck without replacement, the outcome of the

second event depends on the outcome of the first event (the first card being

drawn)

Conditional Probability:

The probability of an event occurring given that another event has already

occurred.

Calculating the Probability of Two Events

Ex1. A computer manufacturer knows that, in a box of 100 computer chips, 3 will be

defective. Jocelyn will draw 2 chips, at random, from a box of 100 chips. Determine the

probability that Jocelyn will draw 2 defective chips.


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Ex2. Determine the probability of drawing two red cards from a standard 52-card deck

assuming the cards are not replaced between drawing.

Notation for Conditional Probability

The notation 𝑃(𝐵|𝐴) is the notation for a conditional probability. It is read “The probability that

event B will occur, given that event A has already occurred”

Since we know that:

𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵|𝐴)

We can rearrange it such that:

𝑃(𝐵|𝐴) =𝑃(𝐴 ∩ 𝐵)

𝑃(𝐴)


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Calculating the Conditional Probability of a Pair of Dependent Events

Ex1. Nathan asks Landon to choose a number between 1 and 40 and then say one fact

about the number. Landon says the number he chose in a multiple of 4. Determine the

probability that the number is also a multiple of 6.

Ex2. Landon then asks Nathan to choose a number from 1 to 40. Nathan told Landon

that it was a multiple of 5. Determine the probability that this number is also a multiple

of 4.


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Solving a Conditional Probability Problem

Ex1. 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.

Ex2. Determine, to the nearest percentage, the probability that any Canadian you met in

the month would have a cellphone that is not a smartphone.


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Making Predictions that Involve Dependent Events

Ex1. Abby is the coach of a junior ultimate (Frisbee) team. Based on the team’s record, it

has a 60% chance of winning on calm days and a 70% chance of winning on windy days.

Tomorrow, there is a 40% chance of high winds. There are no ties in ultimate. What is

the probability that Abby’s team will win tomorrow.

Hint: A tree diagram is very helpful in this situation.

Ex2. Nick’s team has a 75% chance of winning on calm days and a 40% chance of

winning on windy days. For their next game, the forecast predicts a 30% chance of high

winds. Determine the probability that Nick’s team will win their next game.


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Ex3. There are 5 sour cream, 7 chocolate, and 8 honey dip timbits in a box. Determine

the probability that the two timbits that you select at random will be the same flavour.

Practice Problems:

5,6,7,8,9,10,11,13,14,15,16,17,18,19,20 pg 189-191

SUMMERY

If the probability of one event depends on the probability of another event, then

these events are called dependent events. For example, drawing a heart from a

standard deck of 52 playing cards then drawing another heart from the same deck

without replacing the first card are dependent events.

If event B depends on event A occurring, then the conditional probability that

event B will occur, given 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)

A tree diagram is often useful for modelling problems that involve depended events

Drawing an item and then drawing another item, without replacing the first item,

results in a pair of dependent events.


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Section 3.6: Independent Events

Terminology:

Independent Events:

Events whose outcomes are not affected by each other.

Ex. Flipping a coin and rolling a dice

Determine Probabilities of Independent Events

Ex1. Mokhtar and Chantelle 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 to 10 points wins.

Determine the probability that Mokhtar will get 1, 3, or 0 points on his first turn.


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Solving a Problem that Involves Independent Events Using Graphic Organizers

Ex1. 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

a grand prize, and the second draw will be for the consolidation 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 percentage, that

he will win at least one prize.

Practice Problems:

1, 7,8,9,10,11,12,13 pg 198-200

Chapter 3: Probability

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