2. Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds

3. Copyright © 2011 Pearson Education, Inc. Slide 7-3 Unit 7B Combining Probabilities

4. 7-B Copyright © 2011 Pearson Education, Inc. Slide 7-4 Two events are independent if the outcome of one does not affect the probability of the other event. If two independent events A and B have individual probabilities P(A) and P(B), the probability that A and B occur together is P(A and B) = P(A) P(B). This principle can be extended to any number of independent events. And Probability: Independent Events

5. 7-B Copyright © 2011 Pearson Education, Inc. Slide 7-5 Two events are dependent if the outcome of one affects the probability of the other event. The probability that dependent events A and B occur together is P(A and B) = P(A) P(B given A) where P(B given A) is the probability of event B given the occurrence of event A. This principle can be extended to any number of dependent events. And Probability: Dependent Events

6. 7-B Copyright © 2011 Pearson Education, Inc. Slide 7-6 Two events are non- overlapping if they cannot occur together, like the outcome of a coin toss, as shown to the right. For non-overlapping events A and B, the probability that either A or B occurs is shown below. P(A or B) = P(A) + P(B) This principle can be extended to any number of non-overlapping events. Either/Or Probabilities: Non-Overlapping Events

7. 7-B Copyright © 2011 Pearson Education, Inc. Slide 7-7 Two events are overlapping if they can occur together, like the outcome of picking a queen or a club, as shown to the right. For overlapping events A and B, the probability that either A or B occurs is shown below. P(A or B) = P(A) + P(B) – P(A and B) This principle can be extended to any number of overlapping events. Either/Or Probabilities: Overlapping Events

8. 7-B Copyright © 2011 Pearson Education, Inc. Slide 7-8 What is the probability of rolling either a 3 or a 4 on a single six-sided die? These are non-overlapping events. P(A or B) = P(A) + P(B) P(3 or 4) = P(3) + P(4) = 1/6 + 1/6 = 2/6 = 1/3 What is the probability that in a standard shuffled deck of cards you will draw a 5 or a spade? These are overlapping events. P(A or B) = P(A) + P(B) – P(A and B) P(5 or spade) = P(5) + P(spade) – P(5 and spade) = 4/52 + 13/52 – 1/52 = 16/52 = 4/13 Examples

9. 7-B Copyright © 2011 Pearson Education, Inc. Slide 7-9 The At Least Once Rule (For Independent Events) Suppose the probability of an event A occurring in one trial is P(A). If all trials are independent, the probability that event A occurs at least once in n trials is shown below. P(at least one event A in n trials) = 1 – P(not event A in n trials) = 1 – [P(not A in one trial)] n