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**EQ: What are compound events?**

Warm Up One card is drawn from the deck. Find each probability. 1. selecting a two 2. selecting a face card

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**EQ: What are 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.

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**EQ: What are compound events?**

Recall that the union symbol means “or.” Remember!

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**Example 1A: Finding Probabilities of Mutually Exclusive Events**

EQ: What are 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 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**

EQ: What are 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 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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**EQ: What are 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. Each student can vote only once.

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**EQ: What are 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? P(Kline Vila) = P(Kline) + P(Vila) = 20% + 55% = 75%

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**EQ: What are compound events?**

A card is drawn from a deck of cards. Events E1, E2, E3, E4 and E5 are defined as follows: E1: Getting an 8 E2: Getting a king E3: Getting a face card E4: Getting an ace E5: Getting a heart

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**EQ: What are compound events?**

A card is drawn from a deck of cards. Events E1, E2, E3, E4 and E5 are defined as follows: E1: Getting an 8 E2: Getting a king E3: Getting a face card E4: Getting an ace E5: Getting a heart a) Are events E1 and E2 mutually exclusive? b) Are events E2 and E3 mutually exclusive? c) Are events E3 and E4 mutually exclusive? d) Are events E4 and E5 mutually exclusive? e) Are events E5 and E1 mutually exclusive?

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**EQ: What are compound events?**

Two dice are rolled. We define events E1, E2, E3 and E4 as follows E1: Getting a sum equal to 10 E2: Getting a double E3: Getting a sum less than 4 E4: Getting a sum less to 7

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**EQ: What are compound events?**

Two dice are rolled. We define events E1, E2, E3 and E4 as follows E1: Getting a sum equal to 10 E2: Getting a double E3: Getting a sum less than 4 E4: Getting a sum less to 7 a) Are events E1 and E2 mutually exclusive? b) Are events E2 and E3 mutually exclusive? c) Are events E3 and E4 mutually exclusive? d) Are events E4 and E1 mutually exclusive?

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**EQ: What are 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.

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**EQ: What are 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 – 1 = 5. The concept of subtracting the outcomes that are counted twice leads to the following probability formula.

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**EQ: What are compound events?**

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**EQ: What are compound events?**

Recall that the intersection symbol means “and.” Remember!

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**Example 2A: Finding Probabilities of Compound Events**

EQ: What are compound events? 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**

EQ: What are 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 P(odd or >2) = P(odd) + P(>2) – P(odd and >2) There are 2 outcomes where the number is odd and greater than 2.

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**EQ: What are 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 P(king or heart) = P(king) + P(heart) – P(king and heart)

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**EQ: What are 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) P(red or face) = P(red) + P(face) – P(red and face)

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**EQ: What are 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?

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**EQ: What are 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.

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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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**EQ: What are 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?

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**Check It Out! Example 3 Continued**

EQ: What are compound events? 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. hair styling manicure 160 customers 63 28 33

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**Check It Out! Example 3 Continued**

EQ: What are compound events? 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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**EQ: What are compound events?**

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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**EQ: What are 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) Use the complement.

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**EQ: What are compound events?**

Example 4 Continued P(at least 2 students choose same) = 1 – ≈ The probability that at least 2 students choose the same butterfly is about , or 92.31%.

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**EQ: What are 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. P(two customers bought same earrings) = 1 – P(all choose different) Use the complement.

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**Check It Out! Example 4 Continued**

EQ: What are compound events? Check It Out! Example 4 Continued P(at least 2 choose the same) 1 – The probability that at least 2 customers buy the same style is about , or 15.24%.

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**EQ: What are 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? A card can have only one suit.

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**EQ: What are 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?

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**EQ: What are 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?

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