1. Chapter 11 Sequences, Induction, and Probability Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 11.7 Probability

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Compute empirical probability. Compute theoretical probability. Find the probability that an event will not occur. Find the probability of one event or a second event occurring. Find the probability of one event and a second event occurring. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Probability Probabilities of events are expressed as numbers ranging from 0 to 1, or 0% to 100%. The closer the probability of a given event is to 1, the more likely it is that the event will occur. The closer the probability of a given event is to 0, the less likely it is that the event will occur.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Empirical Probability

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Empirical Probabilities with Real-World Data The data in the table are based on 100,000 U.S. women, ages 40 to 50, who participated in mammography screening. Find the probability that a woman aged 40 to 50 has a positive mammogram. The probability is 77%.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Empirical Probabilities with Real-World Data The data in the table are based on 100,000 U.S. women, ages 40 to 50, who participated in mammography screening. Among women with breast cancer, find the probability of a positive mammogram. The probability is 90%.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Empirical Probabilities with Real-World Data The data in the table are based on 100,000 U.S. women, ages 40 to 50, who participated in mammography screening. Among women with positive mammograms, find the probability of having breast cancer. The probability is 9.4%.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Theoretical Probability Any occurrence for which the outcome is uncertain is called an experiment. The set of all possible outcomes of an experiment is the sample space of the experiment, denoted by S. An event, denoted by E, is any subcollection, or subset, of a sample space. Theoretical probability applies to situations in which the sample space only contains equally likely outcomes, all of which are known.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Computing Theoretical Probability

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Computing Theoretical Probability A die is rolled. Find the probability of getting a number greater than 4. The sample space of equally likely outcomes is S = {1, 2, 3, 4, 5, 6}. The event of getting a number greater than 4 can be represented by E = {5, 6}. The probability of getting a number greater than 4 is

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 The Probability of an Event Not Occurring The probability that an event E will not occur is equal to 1 minus the probability that it will occur. P(not E) = 1 – P(E)

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: The Probability of an Event Not Occurring If one person is randomly selected from the world population represented by the figure, find the probability that the person does not live in North America. Express the probability as a simplified fraction and as a decimal rounded to the nearest thousandth.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: The Probability of an Event Not Occurring (continued) P(does not live in North America) = 1 – P(lives in North America) The probability that a person does not live in North America is

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Or Probabilities with Mutually Exclusive Events If it is impossible for any two events, A and B, to occur simultaneously, they are said to be mutually exclusive. If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B). Using set notation,

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: The Probability of Either of Two Mutually Exclusive Events Occurring If you roll a single, six-sided die, what is the probability of getting either a 4 or a 5? P(getting either a 4 or a 5) = P(4) + P(5) The probability of getting either a 4 or a 5 is

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Or Probabilities with Events That Are Not Mutually Exclusive If A and B are not mutually exclusive events, then P(A or B) = P(A) + P(B) – P(A and B). Using set notation,

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: An Or Probability with Real-World Data If one person is randomly selected from the population represented in the table, find the probability that the person is married or female. P(married or female) = P(married) + P(female) – P(married and female)

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: An Or Probability with Real-World Data If one person is randomly selected from the population represented in the table, find the probability that the person is divorced or widowed. P(divorced or widowed) = P(divorced) + P(widowed) – P(divorced and widowed)

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 And Probabilities with Independent Events Two events are independent events if the occurrence of either of them has no effect on the probability of the other. If A and B are independent events, then P(A and B)

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: And Probability with Independent Events Find the probability of a family having four boys in a row. P(four boys in a row) The probability of a family having four boys in a row is