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Probability

Probability. What Are the Chances?. Complements and Unions of Events. 13.2. Understand the relationship between the probability of an event and the probability of its complement. Calculate the probability of the union of two events. Complements and Unions of Events. 13.2.

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Probability

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  1. Probability What Are the Chances?

  2. Complements and Unions of Events 13.2 • Understand the relationship between the probability of an event and the probability of its complement. • Calculate the probability of the union of two events.

  3. Complements and Unions of Events 13.2 • Use complement and union formulas to compute the probability of an event.

  4. Complements of Events

  5. Complements of Events • Example: The graph shows the party affiliation of a group of voters. If we randomly select a person from this group, what is the probability that the person has a party affiliation? (continued on next slide)

  6. Complements of Events • Solution: Let A be the event that the person we select has some party affiliation. It is simpler to calculate the probability of A'. Since 23.7% have no party affiliation, P(A) = 1 – P(A') = 1 – 0.237 = 0.763.

  7. Complements of Events • Example:If the probability that your Blue-Ray player breaks down before the extended warranty expires is 0.015, what is the probability that the player will not break down before the warranty expires? (continued on next slide)

  8. Complements of Events • Example:If the probability that your Blue-Ray player breaks down before the extended warranty expires is 0.015, what is the probability that the player will not break down before the warranty expires? • Solution: 1 - .015 = .985

  9. Unions of Events

  10. Unions of Events • Example: If we select a single card from a standard 52-card deck, what is the probability that we draw either a heart or a face card? (continued on next slide)

  11. Unions of Events • Example: If we select a single card from a standard 52-card deck, what is the probability that we draw either a heart or a face card? • Solution: Let H be the event “draw a heart” and F be the event “draw a face card.” We • are looking for P(H U F). (continued on next slide)

  12. Unions of Events There are 13 hearts, 12 face cards, and 3 cards that are both hearts and face cards.

  13. Unions of Events • Example: If we select a single card from a standard 52-card deck, what is the probability that we draw either a five or a red card? (continued on next slide)

  14. Unions of Events • Example: If we select a single card from a standard 52-card deck, what is the probability that we draw either a five or a red card? • Solution: • P(5) = • P(Red) = • P(5 and Red) (continued on next slide)

  15. Unions of Events • Example:If a pair of dice is rolled, what is the probability that the total showing is either odd or greater than eight? (continued on next slide)

  16. Unions of Events • Example: If a pair of dice is rolled, what is the probability that the total showing is either odd or greater than eight? Solution: • P(> 8) = • P(Odd) = + - = = • P(>8 & Odd) = (continued on next slide)

  17. Unions of Events

  18. Unions of Events • Example: A survey found that 35% of a group of people were concerned with improving their cardiovascular fitness and 55% wanted to lose weight. Also, 70% are concerned with either improving their cardiovascular fitness or losing weight. If a person is randomly selected from the group, what is the probability that the person is concerned with both improving cardiovascular fitness and losing weight? (continued on next slide)

  19. Unions of Events • Solution: Let C be the event “the person wants to improve cardiovascular fitness” and W be the event “the person wishes to lose weight.” We need to find P(C ∩W). • We have P(C) = 0.35, P(W) = 0.55, and P(C U W) = 0.70.

  20. Combining Complement and Union Formulas • A survey of consumers shows the amount of time they spend shopping on the Internet per month compared to their annual income (see below). If we select a consumer randomly, what is the probability that the consumer neither shops on the Internet 10 or more hours per month nor has an annual income above $60,000? (continued on next slide)

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