Compound Events

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11-4
Compound Events
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up One card is drawn from the deck. Find
each probability. 1. selecting a two 2.
selecting a face card Two cards are drawn from
the deck. Find each probability. 3. selecting
two kings when the first card is replaced. 4.
selecting two hearts when the first card is not
replaced.
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Objectives
Find the probability of mutually exclusive
events. Find the probability of inclusive events.
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Vocabulary
simple event compound event mutually exclusive
events inclusive events
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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.
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Example 1A Finding Probabilities of Mutually
Exclusive Events
A group of students is donating blood during a
blood drive. A student has a probability of
having type O blood and a probability of
having type A blood.
Explain why the events type O and type A
blood are mutually exclusive.
A person can only have one blood type.
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Example 1B Finding Probabilities of Mutually
Exclusive Events
A group of students is donating blood during a
blood drive. A student has a probability of
having type O blood and a probability of
having type A blood.
What is the probability that a student has type O
or type A blood?
P(type O ? type A) P(type O) P(type A)
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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.
Each student can vote only once.
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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?
P(Kline ? Vila) P(Kline) P(Vila)
20 55 75
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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.
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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.
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Example 2A Finding Probabilities of Compound
Events
Find the probability on a number cube.
rolling a 4 or an even number
P(4 or even) P(4) P(even) P(4 and even)
4 is also an even number.
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Example 2B Finding Probabilities of Compound
Events
Find the probability on a number cube.
rolling an odd number or a number greater than 2
P(odd or gt2) P(odd) P(gt2) P(odd and gt2)
There are 2 outcomes where the number is odd and
greater than 2.
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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
P(king or heart) P(king) P(heart) P(king
and heart)
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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)
P(red or face) P(red) P(face) P(red and
face)
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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?
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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.
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Example 3 Continued
Step 2 Find the number in the overlapping
region. Subtract 215 from 630. This is the number
of senior paper readers, 415.
Step 3 Find the probability.
P(senior ? reads paper)
P(senior) P(reads paper) P(senior ? reads
paper)
The probability that the student was a senior or
read the daily paper is about 67.6.
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Example 3 Continued
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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?
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Check It Out! Example 3 Continued
Step 1 Use a Venn diagram. Label as much
information as you know. Having a hair styling
and a manicure are inclusive events.
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Check It Out! Example 3 Continued
Step 2 Find the number in the overlapping
region. Subtract 28 from 61. This is the number
of hair stylings and manicures, 33.
Step 3 Find the probability.
P(hair ? manicure) P(hair) P(manicure)
P(hair ? manicure)
The probability that a customer had a hair
styling or manicure is 77.5.
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Recall from Lesson 11-2 that the complement of an
event with probability p, all outcomes that are
not in the event, has a probability of 1 p. You
can use the complement to find the probability of
a compound event.
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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)
Use the complement.
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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.
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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.
P(two customers bought same earrings) 1
P(all choose different)
Use the complement.
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Check It Out! Example 4 Continued
P(at least 2 choose the same) ? 1 0.8476 ?
0.1524
The probability that at least 2 customers buy the
same style is about 0.1524, or 15.24.
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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?
A card can have only one suit.
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Lesson Quiz Part II
The numbers 19 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?
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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?
? 0.5391