1. Probability Probability Day 3 Introduction to Probability Probability of Independent Events

2. Independent Practice/ HW Look at examples on pages 14-15 Do pg. 16-20 ( 17 problems total) http://www.math.harvard.edu/~knil l/mathmovies/swf/21_newton.html

3. Opening Routine

4. Definitions The Probability of an Event, E: Consider a pair of Dice Each of the Outcomes in the Sample Space are random and equally likely to occur. e.g. P( ) = (There are 2 ways to get one 6 and the other 4) P(E) = Number of Event Outcomes Total Number of Possible Outcomes in S

5. Three Types of Probability Theoretical Experimental Subjective

6. Definitions There are three types of probability 1. Theoretical Probability Theoretical probability is used when each outcome in a sample space is equally likely to occur. P(E) = Number of Event Outcomes Total Number of Possible Outcomes in S The Ultimate probability formula

7. Definitions The second type of probability: 2. Experimental Probability Experimental probability is based upon observations obtained from probability experiments. The experimental probability of an event E is the relative frequency of event E P(E) = Number of Event Occurrences Total Number of Observations

8. Definitions The third type of probability: 3. Subjective Probability Subjective probability is a probability measure resulting from intuition, educated guesses, and estimates. Therefore, there is no formula to calculate it. Usually found by consulting an expert.

10. Definitions Law of Large Numbers. As an experiment is repeated over and over, the experimental probability of an event approaches the theoretical probability of the event. The greater the number of trials the more likely the experimental probability of an event will equal its theoretical probability.

11. 11 Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F.

12. The Multiplication Rule If events A and B are independent, then the probability of two events, A and B occurring in a sequence (or simultaneously) is: This rule can extend to any number of independent events. Two events are independent if the occurrence of the first event does not affect the probability of the occurrence of the second event. (More on this later)

13. Example: Two Dice A: roll a 3 with a red die B: roll a 2 with a green die What is the probability of A and B occuring? P(A ∩ B) = P(A) * P(B) = (1/6) * (1/6) = 1/36 P(A ∩ B) = P(A) * P(B) = (1/6) * (1/6) = 1/36

14. Mutually Exclusive Two events A and B are mutually exclusive if and only if: In a Venn diagram this means that event A is disjoint from event B. A and B are M.E. A and B are not M.E.

15. The Addition Rule The probability that at least one of the events A or B will occur, P(A or B), is given by: If events A and B are mutually exclusive, then the addition rule is simplified to: This simplified rule can be extended to any number of mutually exclusive events.

16. Copy… The data in the table are based on 100,000 U.S. women, ages 40 to 50, who participated in mammography screening.

17. 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 7.7%.

18. 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%.

19. 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%.

20. 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)

21. 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.

22. 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 0.921 or 92.1%

23. 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).

24. 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

25. 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,

26. 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 female. P(married or female) = P(married) + P(female) – P(married and female)

27. 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)