1. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. Probability part 1 - review Chapter 7

2. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 2 The Relative Frequency Interpretation of Probability In situations that we can imagine repeating many times, we define the probability of a specific outcome as the proportion of times it would occur over the long run -- called the relative frequency of that particular outcome. Found in two ways: 1.Use a (mathematical) model and calculate probability (e.g. basic Mendelian genetics) 2.Estimate from collected data

3. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 3 Ways to express the relative frequency of lost luggage: The proportion of passengers who lose their luggage is 1/176 or about.006. About 0.6% of passengers lose their luggage. The probability that a randomly selected passenger will lose his/her luggage is about.006. The probability that you will lose your luggage is about.006. Accuracy? MARGIN OF ERROR: Example 7.4 The Probability of Lost Luggage “1 in 176 passengers on U.S. airline carriers will temporarily lose their luggage.”

4. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 4 The Personal Probability Interpretation Personal probability of an event = the degree to which a given individual believes the event will happen. Sometimes subjective probability used because the degree of belief may be different for each individual.

5. compliments mutually exclusive (disjoint): they do not contain any of the same outcomes independent: two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs dependent events Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 5 P(A) + P( A C ) = 1

6. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 6 Conditional Probabilities Conditional probability of the event B, given that the event A occurs, is the long-run relative frequency with which event B occurs when circumstances are such that A also occurs; written as P(B|A). P(B) = unconditional probability event B occurs. P(B|A) = “probability of B given A” = conditional probability event B occurs given that we know A has occurred or will occur.

7. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 7 Rule 4 (conditional probability): P(B|A) = P(A and B)/P(A) P(A|B) = P(A and B)/P(B) Determining a Conditional Probability

8. Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 8

9. 9 A sample is drawn with replacement if individuals are returned to the eligible pool for each selection. A sample is drawn without replacement if sampled individuals are not eligible for subsequent selection. Sampling with and without Replacement Example: Consider randomly selecting two left-handed students from a class of 30, where 3 students are left-handed.

10. Example: Suppose that events A and B are mutually exclusive with probability P(A)=1/2 and probability P(B)=1/3. (a)Are A and B independent? (b)Are A and B complementary events? 10

11. Example: When a fair die is tossed, each of the six sides (numbers 1 through 6) are equally likely to land face up. Two fair dice, one red and one green, are tossed. (a). A = red die is a 3; B = red die is a 6 (b). A = red die is a 3; B = green die is a 6 (c). A = red die and green die sum to 4; B = red die is a 3. (d). A = red die and green die sum to 4; B = red die is a 4. In which case(s) are A and B disjoint? Independent? 11

12. Example: Suppose you toss a fair coin 6 times. Which of the following sequences is the most likely? HTHTTH HHHTTT HHHHTH 12

13. Example 7.14: Suppose that there has been a crime and it is known that that the criminal is a person within a population of 6,000,000. Further, suppose that it is known that in this population only about one person in a million has a DNA type that matches the DNA found at the crime scene, so assume that there are six people in the population whose DNA would match. An individual in custody has matching DNA. What is the probability that he is innocent? 13

14. Los Angeles Times (August 24, 1987): Several studies of sexual partners of people infected with [HIV] show that a single act of unprotected vaginal intercourse has a surprisingly low risk of infecting the uninfected partner – perhaps one in 100 to one in 1000. For an average, consider the risk to be one in 500. If there are 100 acts of intercourse with an infected partner, the odds of infection increase to one in five. Statistically, 500 acts of intercourse with one infected partner or 100 acts of intercourse with five partners lead to a 100% probability of infection (statistically, not necessarily in reality). 14