Two cards are drawn from a deck of 52.
Determine whether the events are
independent or dependent. Find the
indicated probability.
A. selecting two face cards when the first
card is replaced
B. selecting two face cards when the first
card is not replaced
Warm up
This unit
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Fundamental counting principle
factorial
Permutations
Combinations
probability
complement
experimental probability
theoretical probability
independent events
dependent events
conditional probability
Fundamental Counting
Principle
Lets start with a simple example.
A student is to roll a die and flip a coin.
How many possible outcomes will there be?
Example 2A: Finding Permutations
How many ways can a student government select a
president, vice president, secretary, and treasurer
from a group of 6 people?
Example 3: Application
There are 12 different-colored cubes in a bag. How
many ways can Randall draw a set of 4 cubes from
the bag?
Check It Out! Example 1a
A red number cube and a blue
number cube are rolled. If all
numbers are equally likely,
what is the probability of the
event?
The sum is 6.
Check It Out! Example 3
A DJ randomly selects 2 of 8 ads to play
before her show. Two of the ads are by a local
retailer. What is the probability that she will
play both of the local retailer’s ads before her
show?
Check It Out! Example 5b
The table shows the results of choosing one card from a deck of cards,
recording the suit, and then replacing the card.
Find the experimental probability of choosing a card that is not a club.
1. Find the probability of rolling a number
greater than 2 and then rolling a
multiple of 3 when a number cube is
rolled twice.
A drawer contains 8 blue socks, 8 black
socks, and 4 white socks. Socks are
picked at random. Explain why the
events picking a blue sock and then
another blue sock are dependent. Then
find the probability.
7-5 Compound Events
A simple event is an event that describes a single
outcome. A compound event is an event made up of
two or more simple events. Mutually exclusive
events are events that cannot both occur in the same
trial of an experiment. Rolling a 1 and rolling a 2 on
the same roll of a number cube are mutually exclusive
events.
Holt McDougal Algebra 2
Remember!
Recall that the union symbol  means “or.”
7-5 Compound Events
Example 1A: Finding Probabilities of Mutually
Exclusive Events
A group of students is donating blood during a
blood drive. A student has a
having type O blood and a
probability of
probability of
having type A blood.
Explain why the events “type O” and “type A”
blood are mutually exclusive.
Holt McDougal Algebra 2
7-5 Compound Events
Example 1B: Finding Probabilities of Mutually
Exclusive Events
A group of students is donating blood during a
blood drive. A student has a
having type O blood and a
probability of
probability of
having type A blood.
What is the probability that a student has type O
or type A blood?
Holt McDougal Algebra 2
7-5 Compound Events
Check It Out! Example 1a
Each student cast one vote for senior class
president. Of the students, 25% voted for Hunt,
20% for Kline, and 55% for Vila. A student from
the senior class is selected at random.
Explain why the events “voted for Hunt,” “voted
for Kline,” and “voted for Vila” are mutually
exclusive.
Holt McDougal Algebra 2
7-5 Compound Events
Check It Out! Example 1b
Each student cast one vote for senior class
president. Of the students, 25% voted for Hunt,
20% for Kline, and 55% for Vila. A student from
the senior class is selected at random.
What is the probability that a student voted for
Kline or Vila?
Holt McDougal Algebra 2
7-5 Compound Events
Inclusive events are events that have one or more
outcomes in common. When you roll a number cube, the
outcomes “rolling an even number” and “rolling a prime
number” are not mutually exclusive. The number 2 is both
prime and even, so the events are inclusive.
Holt McDougal Algebra 2
7-5 Compound Events
There are 3 ways to roll an even number, {2, 4, 6}.
There are 3 ways to roll a prime number, {2, 3, 5}.
The outcome “2” is counted twice when outcomes are
added (3 + 3) . The actual number of ways to roll an
even number or a prime is 3 + 3 – 1 = 5. The concept
of subtracting the outcomes that are counted twice
leads to the following probability formula.
Holt McDougal Algebra 2
7-5 Compound Events
Remember!
Recall that the intersection symbol  means
“and.”
Holt McDougal Algebra 2
7-5 Compound Events
Example 2A: Finding Probabilities of Compound
Events
Find the probability on a number cube.
rolling a 4 or an even number
Holt McDougal Algebra 2
7-5 Compound Events
Example 2B: Finding Probabilities of Compound
Events
Find the probability on a number cube.
rolling an odd number or a number greater than 2
Holt McDougal Algebra 2
7-5 Compound Events
Check It Out! Example 2a
A card is drawn from a deck of 52. Find the
probability of each.
drawing a king or a heart
Holt McDougal Algebra 2
7-5 Compound Events
Check It Out! Example 2b
A card is drawn from a deck of 52. Find the
probability of each.
drawing a red card (hearts or diamonds) or a
face card (jack, queen, or king)
Holt McDougal Algebra 2
7-5 Compound Events
Example 3: Application
Of 1560 students surveyed, 840 were seniors and
630 read a daily paper. The rest of the students
were juniors. Only 215 of the paper readers were
juniors. What is the probability that a student
was a senior or read a daily paper?
Holt McDougal Algebra 2
7-5 Compound Events
Example 3 Continued
Step 1 Use a Venn diagram.
Label as much information as you know. Being a
senior and reading the paper are inclusive events.
Holt McDougal Algebra 2
7-5 Compound Events
Example 3 Continued
Step 2 Find the number in the overlapping region.
Step 3 Find the probability.
Holt McDougal Algebra 2
7-5 Compound Events
Check It Out! Example 3
Of 160 beauty spa customers, 96 had a hair
styling and 61 had a manicure. There were 28
customers who had only a manicure. What is the
probability that a customer had a hair styling or
a manicure?
Holt McDougal Algebra 2
7-5 Compound Events
Example 4 Application
Each of 6 students randomly chooses a butterfly
from a list of 8 types. What is the probability
that at least 2 students choose the same
butterfly?
P(at least 2 students choose same) = 1 – P(all choose different)
Holt McDougal Algebra 2
7-5 Compound Events
Example 4 Continued
P(at least 2 students choose same) = 1 – 0.0769 ≈ 0.9231
The probability that at least 2 students choose the
same butterfly is about 0.9231, or 92.31%.
Holt McDougal Algebra 2
7-5 Compound Events
Check It Out! Example 4
In one day, 5 different customers bought
earrings from the same jewelry store. The store
offers 62 different styles. Find the probability
that at least 2 customers bought the same style.
Holt McDougal Algebra 2
7-5 Compound Events
Lesson Quiz: Part I
You have a deck of 52 cards.
1. Explain why the events “choosing a club” and
“choosing a heart” are mutually exclusive.
2. What is the probability of choosing a club or a
heart?
Holt McDougal Algebra 2
7-5 Compound Events
Lesson Quiz: Part II
The numbers 1–9 are written on cards and
placed in a bag. Find each probability.
3. choosing a multiple of 3 or an even number
4. choosing a multiple of 4 or an even number
5. Of 570 people, 365 were male and 368 had
brown hair. Of those with brown hair, 108 were
female. What is the probability that a person
was male or had brown hair?
Holt McDougal Algebra 2
7-5 Compound Events
Lesson Quiz: Part III
6. Each of 4 students randomly chooses a pen
from 9 styles. What is the probability that at
least 2 students choose the same style?
Holt McDougal Algebra 2