Understanding Compound Events in Probability - Holt McDougal Algebra 2 Lesson

1. 7-5 Compound Events Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

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

3. Objectives Find the probability of mutually exclusive events. Find the probability of inclusive events.

4. Vocabulary simple event compound event mutually exclusive events inclusive events

5. A simple eventis an event that describes a single outcome. A compound eventis an event made up of two or more simple events. Mutually exclusiveeventsare 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.

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

7. 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.

8. 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)

9. 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.

10. 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%

11. Inclusive eventsare 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.

12. 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.

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

15. 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.

16. 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.

17. 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)

18. 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)

19. 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?

20. 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.

21. 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%.

22. Example 3 Continued

23. 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?

24. 160 customers 63 33 28 hair styling manicure 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.

25. 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%.

26. 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.

27. 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.

28. 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%.

29. 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.

30. 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%.

31. 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.

32. 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?

33. 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