Probability

1. Probability Conditional Probability and Independence

2. Importance of the Sample Space The probability of an event depends on the sample space in question. The sample space is critical to determining the probability. In certain types of probabilistic situations, the entire sample space is not utilized � only a portion is needed.

3. Rolling a Single Die Consider rolling a single die. List the sample space. What is the probability of rolling a 5? What is the probability of rolling a 5 given that an odd number has been rolled?

5. Example: Conditional Probability Twenty percent of the employees of Acme Steel Company are college graduates. Of all its employees, 25% earn more than $50,000 per year, and 15% are college graduates earning more than $50,000. What is the probability that an employee selected at random earns more than $50,000 per year, given that he or she is a college graduate?

6. Conditional Probability � Equally Likely Outcomes

7. Example: Conditional Probability A sample of two balls are selected from an urn containing 8 white balls and 2 green balls. What is the probability that the second ball selected is white given that the first ball selected was white?

8. Product Rule Product Rule If Pr(F) ? 0, Pr(E n F) = Pr(F) ? Pr(E | F). The product rule can be extended to three events. Pr(E1 n E2 n E3) = Pr(E1) ? Pr(E2 | E1) ? Pr(E3| E1 n E2)

9. Example: Product Rule A sequence of two playing cards is drawn at random (without replacement) from a standard deck of 52 cards. What is the probability that the first card is red and the second is black? Let F = "the first card is red," and E = "the second card is black."

10. Independent Events Events are said to be independent if the probability of one event does not affect the likelihood of occurrence of the other event(s). A collection of events is said to be independent if for each collection of events chosen from them, the probability that all the events occur equals the product of the probabilities that each occurs.

12. Example: Independence Let an experiment consist of observing the results of drawing two consecutive cards from a 52-card deck. Let E = "second card is black" and F = "first card is red". Are these two events independent?

15. Example 1 The proportion of individuals in a certain city earning more than $35,000 per year is .25. The proportion of individuals earning more than $35,000 and having a college degree is .10. Suppose that a person is randomly chosen and he turns out to be earning more than $35,000. What is the probability that he is a college graduate?

16. Example 2 A stereo system contains 50 transistors. The probability that a given transistor will fail in 100,000 hours of use is .0005. Assume that the failures of the various transistors are independent of one another. What is the probability that no transistor will fail during the first 100,000 hours of use?

17. Example 3 Let E and F be events with P(E) = .3, P(F) = .6, and P(E ? F) = .7 . Find: a.) P(E n F) b.) P(E | F) c.) P(F | E) d.) P(E? n F) e.) P(E ? | F)

18. Example 4 Of the students at a certain college, 50% regularly attend the football games, 30% are first-year students, and 40% are upper-class students who do not regularly attend football games. Suppose that a student is selected at random. a.) What is the probability that the person both is a first-year student and regularly attends football games?

19. Example 4 (continued) Of the students at a certain college, 50% regularly attend the football games, 30% are first-year students, and 40% are upper-class students who do not regularly attend football games. Suppose that a student is selected at random. b.) What is the probability that the person regularly attends football games given that he is a first-year student? c.) What is the probability that the person is a first-year student given that he regularly attends football games?

20. Example 5 Two poker chips are selected at random from an bag containing two white chips and three red chips. What is the probability that both chips are white given that at least one of them is white?

21. Example 6 Out of 250 students interviewed at a community college, 90 were taking mathematics but not computer science, 160 were taking mathematics, and 50 were taking neither mathematics nor computer science. Find the probability that a student chosen at random was: a.) taking just computer science.

22. Example 6 Out of 250 students interviewed at a community college, 90 were taking mathematics but not computer science, 160 were taking mathematics, and 50 were taking neither mathematics nor computer science. Find the probability that a student chosen at random was: b.) taking mathematics or computer science, but not both. c.) taking mathematics, given that the student was taking computer science. d.) taking computer science, given that the student was not taking mathematics.

23. Example 7 Find the probability that a student selected at random is: a.) a senior. b.) working full-time. c.) working part-time, given that the student is a first- year student. d.) a junior or senior, given that the student does not work.

24. Example 8 A bag contains eight purple marbles, six blue marbles, and 12 red marbles. Four marbles are selected from the bag. a.) What is the probability that they are all purple? b.) What is the probability that they are all the same color?