Understanding Mutually Exclusive and Inclusive Events

1. 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.

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

3. 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.

4. 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)

5. 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.

6. 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%

7. 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.

8. 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.

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

11. 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.

12. 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.

13. 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)

14. 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)

15. Example 3: Application Of 1560 students surveyed, 840 were seniors. 630 of total students 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?

16. 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.

17. 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%.

18. Example 3 Continued

19. Check It Out! Example 3 Of 157 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?

20. 157 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.

21. 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 78.98%.

22. Recall from Lesson 7-1 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.

23. Example 4 Application • Two dice are tossed. Find the probability of each event. • Their sum is seven • Their sum is not seven P (sum is 7) = 6/36 = 1/6 P(sum is not 7) = 1 – P(sum is 7) = 1 – 1/6 = 5/6 Use the complement.