chapter 5 probability 5.2 the addition rule; complements

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Chapter 5 Probability 5.2 The Addition Rule; Complements

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Page 1: Chapter 5 Probability 5.2 The Addition Rule; Complements

Chapter 5Probability

5.2

The Addition Rule; Complements

Page 2: Chapter 5 Probability 5.2 The Addition Rule; Complements

Let E and F be two events.

E and F is the event consisting of simple events that belong to both E and F.

E or F is the event consisting of simple events that belong to either E or F or both.

Page 3: Chapter 5 Probability 5.2 The Addition Rule; Complements

EXAMPLE Illustrating the Addition Rule

Suppose that a pair of dice are thrown. Let E = “the first die is a two” and let F = “the sum of the dice is less than or equal to 5”. Find P(E or F) directly by counting the number of ways E or F could occur and dividing this result by the number of possible outcomes.

Page 4: Chapter 5 Probability 5.2 The Addition Rule; Complements

Addition Rule

For any two events E and F,

P(E or F) = P(E) + P(F) – P(E and F)

Page 5: Chapter 5 Probability 5.2 The Addition Rule; Complements

EXAMPLE The Addition Rule

Redo the last example using the Addition Rule.

Page 6: Chapter 5 Probability 5.2 The Addition Rule; Complements

Venn diagrams represent events as circles enclosed in a rectangle. The rectangle represents the sample space and each circle represents an event.

Page 7: Chapter 5 Probability 5.2 The Addition Rule; Complements
Page 8: Chapter 5 Probability 5.2 The Addition Rule; Complements

If events E and F have no simple events in common or cannot occur simultaneously, they are said to be disjoint or mutually exclusive.

Page 9: Chapter 5 Probability 5.2 The Addition Rule; Complements

Addition Rule for Mutually Exclusive EventsAddition Rule for Mutually Exclusive Events

If E and F are mutually exclusive events, then

P(E or F) = P(E) + P(F)

In general, if E, F, G, … are mutually exclusive events, then

P(E or F or G or …) = P(E) + P(F) + P(G) + …

Page 10: Chapter 5 Probability 5.2 The Addition Rule; Complements

Events E and F are Mutually Exclusive

Events E, F and G are Mutually Exclusive

Page 11: Chapter 5 Probability 5.2 The Addition Rule; Complements

EXAMPLE Using the Addition Rule

The following data represent the language spoken at home by age for residents of San Francisco County, CA between the ages of 5 and 64 years.

Source: United States Census Bureau, 2000 Supplementary Survey

Page 12: Chapter 5 Probability 5.2 The Addition Rule; Complements

(a) What is the probability a randomly selected resident of San Francisco County between 5 and 64 years speaks English only at home?

(b) What is the probability a randomly selected resident of San Francisco between 5 and 64 years is 5 - 17 years old?

(c ) What is the probability a randomly selected resident of San Francisco between 5 and 64 years is 5 - 17 years old or speaks English only at home?

Page 13: Chapter 5 Probability 5.2 The Addition Rule; Complements
Page 14: Chapter 5 Probability 5.2 The Addition Rule; Complements
Page 15: Chapter 5 Probability 5.2 The Addition Rule; Complements

EXAMPLE Illustrating the Complement Rule

According to the American Veterinary Medical Association, 31.6% of American households own a dog. What is the probability that a randomly selected household does not own a dog?

Page 16: Chapter 5 Probability 5.2 The Addition Rule; Complements

EXAMPLE Illustrating the Complement Rule

The data on the following page represent the travel time to work for residents of Hartford County, CT.

(a) What is the probability a randomly selected resident has a travel time of 90 or more minutes?

(b) What is the probability a randomly selected resident has a travel time less than 90 minutes?

Page 17: Chapter 5 Probability 5.2 The Addition Rule; Complements

Source: United States Census Bureau, 2000 Supplementary Survey