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Geometry Unit 12 Probability Notes
1
Date Name of Lesson
12.1 Representing Sample Spaces
12.2 Probability and Counting
12.4 Geometric Probability
12.5 Probability and the Multiplication Rule
12.6 Probability and the Addition Rule
12.7 Conditional Probability
12.8 Two-Way Frequency Tables
Geometry Unit 12 Probability Notes
2
12.1 Representing Sample Spaces
Example 1: Represent a Sample Space
One red token and one black token are placed in a
bag. A token is drawn, and the color is recorded. It is
then returned to the bag, and a second draw is made.
Represent the sample space for this experiment by
making an organized list, a table, and a tree diagram.
Create and Organized List
Create a Table
Create a Tree Diagram
Guided Practice 1: Represent a Sample Space
One yellow token and one blue token are placed in a
bag. A token is drawn and the color is recorded. It is
then returned to the bag and a second draw is made.
Choose the correct display of this sample space.
Create and Organized List
Create a Table
Create a Tree Diagram
Example 2:
Multistage Tree Diagrams
CHEF’S SALAD A chef’s salad at a local restaurant
comes with a choice of French, ranch, or blue cheese
dressings and optional toppings of cheese, turkey, and
eggs. Draw a tree diagram to represent the sample
space for salad orders.
Draw a tree diagram with 4 stages.
The sample space is the result of 4 stages.
● Dressing (F, R, or BC)
● Cheese (C or NC)
● Turkey (T or NT)
● Eggs (E or NE)
Geometry Unit 12 Probability Notes
3
Use the Fundamental Counting Principal
CARS New cars are available with a wide selection
of options for the consumer. One option is chosen
from each category shown. How many different cars
could a consumer create in the chosen make and
model?
BICYCLES New bicycles are available with a wide
selection of options for the rider. One option is
chosen from each category shown. How many
different bicycles could a consumer create in the
chosen model?
Geometry Unit 12 Probability Notes
4
12.2 Probability and Counting
Example 1: Find the Union of Events
A set of 12 cards are numbered 1 through 12. You
choose one card at random. Let A be the event that
you choose a multiple of 4. Let B be the event that
you choose a number less than 3.
Find A ∪ B
Find the probability that event A or event B will
occur.
Example 2: Find the Intersection of Events
The Venn diagram shows the employees of a store
and the days of the weekend when they work. Let A
be the event that an employee works Saturday and let
B be the event that an employee works Sunday. An
employee is chosen at random to attend a training
session.
Find A ∩ B What is the probability that the employee who is
chosen works on both Saturday and Sunday?
Geometry Unit 12 Probability Notes
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Example 3: Use Complementary Events
A box contains 35 red marbles and 120 marbles
of other colors. Isaac chooses one of the marbles
without looking. What is the probability that he
does not choose a red marble?
Example 4: Find Probabilities of Events
Some visitors to an amusement park were surveyed to find
out whether they rode the roller coaster or the Ferris
wheel. The Venn diagram represents the results of the
survey. One respondent will be chosen at random to win a
free pass to the park. Find the probability that the winner
will be someone who rode the roller coaster and the Ferris
wheel.
Geometry Unit 12 Probability Notes
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12.4 Geometric Probability
Example 1: Lengths and Geometric Probability
Point Z is chosen at random on AD. Find the
probability that Z is on AB.
Guided Practice 1: Length Probabilities
Point R is chosen at random on LO. Find the
probability that R is on MN.
Example 2:
Halley’s Comet orbits Earth every
76 years. What is the probability that Halley’s Comet
will complete an orbit within the next decade?
Example 2B: Translations in a Coordinate Plane
SUBWAY You are in the underground station
waiting for the next subway car, and are unsure how
long ago the last one left. You do know that the
subway comes every sixteen minutes. What is the
probability that you will get picked up in the next
12 minutes?
Geometry Unit 12 Probability Notes
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Example 3: Area and Geometric Probability
DARTS The targets of a dartboard are formed by
3 concentric circles. If the diameter of the center
circle is 4 inches and the circles are spread
3 inches apart, what is the probability that a
player will throw a dart into the center circle?
Guided Practice 3:
RING TOSS If at a carnival,
you toss a ring and it lands in
the red circle shown below,
then you win a prize. The
diameter of the circle is 4 feet.
If the dimensions of the blue
table are 8 feet by 5 feet, what
is the probability if the ring is
thrown at random that you will
win a prize?
Example 4: Angle Measure Probability
Use the spinner to find
the P (pointer landing
on section 3).
Use the spinner to find
the P (pointer landing
on section 1).
Guided Practice 4:
Use the spinner to find the
P (pointer landing on
section C).
Use the spinner to find the
P (pointer landing on
section E).
Geometry Unit 12 Probability Notes
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12.5 Probability and the Multiplication Rule
Independent Events:
Dependent Events:
Example 1:
Determine whether the event is independent or
dependent. Explain your reasoning.
A. Anna rolls a 6 on one number cube and a 3 on
another cube.
B. A queen is selected from a standard deck of cards and not put back. Then a king is selected.
Guided Practice 1:
Determine whether the event is independent or
dependent. Explain your reasoning.
A. A marble is selected from a bag. It is not put back.
Then a second marble is selected.
B. A marble is selected from a bag. Then a card is
selected from a deck of cards.
Example 2:
A bag contains a white marble, a blue marble, a
yellow marble, and a green marble. Andrew selects
the white marble, replaces it, and then selects the
green marble.
Are these events independent?
Explain using probability.
Geometry Unit 12 Probability Notes
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Guided Notes:
EATING OUT Michelle and Christina are going out to lunch. They put 5 green slips of paper and 6 red slips
of paper into a bag. If a person draws a green slip, she will order a hamburger. If she draws a red slip, she will
order pizza. Michelle will draw first and put her slip back. Then Christina will draw. What is the probability
that both girls draw green slips?
Your Turn:
LABS In Science class, students are drawing marbles out of a bag to determine lab groups. There are 4 red
marbles, 6 green marbles, and 5 yellow marbles left in the bag. Jacinda draws a marble, but not liking the
outcome, she puts it back and draws a second time. What is the probability that each of her 2 draws gives her
a red marble?
Geometry Unit 12 Probability Notes
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GAMES At the school carnival, winners in the ringtoss game are randomly given a prize from a bag that
contains 4 sunglasses, 6 hairbrushes, and 5 key chains.
The first three players all win prizes. Find each probability.
A. P (sunglasses, hairbrush, key chain)
B. P (hairbrush, hairbrush, not a hairbrush)
LABS In Science class, students are again drawing marbles out of a bag to determine lab groups. There are 4
red marbles, 6 green marbles, and
5 yellow marbles. This time Graham draws a marble and does not put his marble back in the bag. Then his
friend Meena draws a marble. What is the probability they both draw green marbles?
Davina’s family will cancel their weekend camping trip if the probability of rain on both Saturday and Sunday
is greater than 10%. According to the weather forecast, there is a 30% chance of rain on Saturday and a 20%
chance of rain on Sunday. Assuming the two events (rain on Saturday and rain on Sunday) are independent,
should Davina’s family cancel the trip? Justify your answer using probability
Find the probability of rain on Saturday and Sunday.
Should Davina’s family cancel their weekend trip?
Geometry Unit 12 Probability Notes
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12.6 Probability and the Addition Rule
Mutual Exclusive Events:
Not Mutually Exclusive Events:
Example 1A
CARDS Han draws one card from a standard deck.
Determine whether drawing an ace or a 9 is mutually
exclusive or not mutually exclusive. Explain your
reasoning.
Example 1B
CARDS Han draws one card from a standard deck.
Determine whether drawing a king or a club is
mutually exclusive or not mutually exclusive. Explain
your reasoning.
Your Turn 1A
A. For a Halloween grab bag, Mrs. Roth has thrown
in 10 caramel candy bars, 15 peanut butter candy
bars, and 5 apples to have a healthy option.
Determine whether drawing a candy bar or an apple
is mutually exclusive or not mutually exclusive.
Your Turn 2A
B. For a Halloween grab bag, Mrs. Roth has thrown
in 10 caramel candy bars, 15 peanut butter candy
bars, and 5 apples to have a healthy option.
Determine whether drawing a candy bar or something
with caramel is mutually exclusive or not mutually
exclusive.
Example 2:
COINS Trevor reaches into a can that contains 30
quarters, 25 dimes, 40 nickels, and 15 pennies. What
is the probability that the first coin he picks is a
quarter or a penny?
Your Turn 2:
MARBLES Hideki collects colored marbles so he
can play with his friends. The local marble store has a
grab bag that has 15 red marbles, 20 blue marbles, 3
yellow marbles and 5 mixed color marbles. If he
reaches into a grab bag and selects a marble, what is
the probability that he selects a red or a mixed color
marble?
Geometry Unit 12 Probability Notes
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Example 3:
ART Use the table below. What is the probability
that Namiko selects an acrylic or a still life?
Your Turn 3:
SPORTS Use the table. What is the probability that
if a high school athlete is selected at random that the
student will be a sophomore or a basketball player?
Geometry Unit 12 Probability Notes
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12.7 Conditional Probability
Example 1
Mr. Monroe is organizing the gym class into two teams for a game. The 20 students randomly draw cards
numbered with consecutive integers from 1 to 20.
• Students who draw odd numbers will be on the Red team.
• Students who draw even numbers will be on the Blue team.
If Monica is on the Blue team, what is the probability that she drew the number 10?
Your Turn 1
Mr. Riley’s class is traveling on a field trip for Science class. There are two busses to take the students to a
chemical laboratory. To organize the trip, 32 students randomly draw cards numbered with consecutive
integers from 1 to 32.
• Students who draw odd numbers will be on the first bus.
• Students who draw even numbers will be on the second bus.
If Yael will ride the second bus, what is the probability that she drew the number 18 or 22?
Example 2
At a fruit stand, 24% of the grape bags have red grapes, 15% have black grapes, and 3% have both red and
black grapes. A customer selects a bag at random.
A. What is the probability that the bag contains red grapes, given that it contains black grapes?
B. What is the probability that the bag contains black grapes, given that it contains red grapes?
Geometry Unit 12 Probability Notes
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Your Turn 2
LABS In Science class, students are drawing marbles out of a bag to determine lab groups. There are 4 red
marbles, 6 green marbles, and 5 yellow marbles left in the bag. Jacinda draws a marble, but not liking the
outcome, she puts it back and draws a second time. What is the probability that each of her 2 draws gives her
a red marble?
Example 3
PETS A survey of Kingston High School students found that 63% of the students had a cat or a dog for a pet.
If two students are chosen at random from a group of 100 students, what is the probability that at least one of
them does not have a cat or a dog for a pet?
Your Turn 3
LABS In Science class, students are again drawing marbles out of a bag to determine lab groups. There are 4
red marbles, 6 green marbles, and 5 yellow marbles. This time Graham draws a marble and does not put his
marble back in the bag. Then his friend Meena draws a marble. What is the probability they both draw green
marbles?
Geometry Unit 12 Probability Notes
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12.8 Two-Way Frequency Tables
Example 1
Valerie asks a random sample of 70 science teachers and math teachers whether they have been to the town’s
planetarium. She finds that 25 science teachers have been to the planetarium and 3 have not, while 20 math
teachers have been to the planetarium and 22 have not. Make a two-way frequency table to organize the data.
Example 2
Joint Frequency:
Marginal Frequency:
A. How many customers at Donna’s Diner paid by credit card? Is the frequency marginal or joint?
B. How many customers paid by cash? Is the frequency marginal or joint?
Geometry Unit 12 Probability Notes
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Example 3
Convert the below table to a table of relative frequencies.
Example 4
Use the below relative frequency table to determine whether paying by cash or credit card is independent of
the restaurant. Explain.
Example 5
Use the relative frequency table above to find the probability that a customer pays by cash, given that he or
she eats at Joe’s Palace.