1. Copyright © Cengage Learning. All rights reserved. Elementary Probability Theory 5

2. Copyright © Cengage Learning. All rights reserved. Section 5.2 Some Probability Rules—Compound Events

3. 3 Focus Points Compute probabilities of general compound events. Compute probabilities involving independent events or mutually exclusive events. Use survey results to compute conditional probabilities.

4. 4 Conditional Probability and Multiplication Rules

5. 5 Independent events

6. 6 Conditional Probability and Multiplication Rules Conditional probability P(A|B) If the events are dependent, then we must take into account the changes in the probability of one event caused by the occurrence of the other event. The notation P(A, given B) denotes the probability that event A will occur given that event B has occurred. This is called a conditional probability.

7. 7 Independent or dependent A = Being female B = Having brown hair A = 1st child is a boy B = 2nd child is a boy Examples of Independent When event A affects event B, they are dependent. A = taking an aspirin each day B = having a heart attack A = studying for a test B = making an A on a test

8. 8 With or Without Replacement Without replacement = dependent What is the probability of drawing three kings without replacement? With replacement = independent What is the probability of drawing three kings with replacement?

9. 9 1.Practicing the piano, and then becoming a concert pianist. 2.Tossing a coin, getting a head, then rolling a die and getting a 6. 3.Selecting a king from a standard deck, replacing it, and then selecting a queen from the deck. 4.Selecting a king from a standard deck, not replacing it, and then selecting a queen from the deck. 5.Parking beside a fire hydrant on Tuesday and getting a parking ticket on Tuesday. 6.A numbered ball between 1 and 40 is selected from a bin, and then a second numbered ball is selected from the remaining balls in the bin. Independent or dependent

10. 10 Conditional Probability and Multiplication Rules We read P(A, given B) as “probability of A given B.” If A and B are dependent events, then P(A)  P(A, given B) because the occurrence of event B has changed the probability that event A will occur. A standard notation for P(A, given B) is P(A | B).

11. 11 The results of responses when a sample of adults in 3 cities was asked if they liked a new juice is: 1. P(Yes) 2. P(Seattle) 3. P(Miami) 4. P (No │given Miami) OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 One of the responses is selected at random. Find: 1. P (No │given Omaha) 2. P (yes │given Omaha) 3. P (Miami │given No) 4. P (yes │ given Seattle)

12. 12 Class practice Did not graduate high school High school graduate Some college College graduate Total Never Married401277906565768526,052 Married, Spouse Present 15122360762700329811108,012 Married, Spouse Absent 18772006143514896,807 Separated1094151512145984,421 Widowed500047542264145613,474 Divorced295170475649376619,413 Total30,05659,18844,13044,805178,179

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