1. Statistics 270 - Lecture 6

2. Last day: Probability rules Today: Conditional probability Suggested problems: Chapter 2: 45, 47, 59, 63, 65

3. Example – Let’s Make a Deal: A contestant is given a choice of three doors of which one contained a prize such as a Car The other two doors contained gag gifts like a chicken or a donkey After the contestant choses an initial door, the host of the show reveals an empty door among the two unchosen doors, and asks the contestant if they would like to switch to the other unchosen door What should the contestant do?

4. Conditional Probability Sometimes interested in in probability of an event, after information regarding another event has been observed The conditional probability of an event A, given that it is known B has occurred is: Called “probability of A given B ”

5. Example – Let’s Make a Deal: A contestant is given a choice of three doors of which one contained a prize such as a Car The other two doors contained gag gifts like a chicken or a donkey After the contestant choses an initial door, the host of the show reveals an empty door among the two unchosen doors, and asks the contestant if they would like to switch to the other unchosen door What should the contestant do?

6. Some Useful Formulas Multiplication Rule: Law of Total Probability: Bayes Theorem:

7. Where do these come from?

8. Example In a region 12% of adults are smokers, 0.8% are smokers with emphysema and 0.2% are non-smokers with emphysema What is the probability that a randomly selected individual has emphysema? Given that the person is a smoker, what is the probability that the person has emphysema?

9. Example From a group of 5 Democrats, 5 Republicans and 5 Independents, a committee of size 3 is to selected What is the probability that each group will be represented on the committee if the first person selected is an Independent?

10. Example Consider a routine diagnostic test for a rare disease Suppose that 0.1% of the population has the disease, and that when the disease is present the probability that the test indicates the disease is present is 0.99 Further suppose that when the disease is not present, the probability that the test indicates the disease is present is 0.10 For the people who test positive, what is the probability they actually have the disease

11. Example (Randomized Response Model) Can design survey using conditional probability to help get honest answer for sensitive questions Want to estimate the probability someone cheats on taxes Questionnaire: 1. Do you cheat on your taxes? 2. Is the second hand on the clock between 12 and 3? YES NO Methodology: Sit alone, flip a coin and if the outcome is heads answer question 1 otherwise answer question 2

12. Several Events Suppose (A 1, A 2, …, A k ) form a partition of the sample space…i.e., they are mutually exclusive and their union equals the sample space Bayes Theorem: suppose (A 1, A 2, …, A k ) form a partition of the sample space

13. Independent Events Two events are independent if: The intuitive meaning is that the outcome of event B does not impact the probability of any outcome of event A Alternate form:

14. Example Flip a coin two times S= A={head observed on first toss} B={head observed on second toss} Are A and B independent?

15. Example Mendel used garden peas in experiments that showed inheritance occurs randomly Seed color can be green or yellow {G,G}=Green otherwise pea is yellow Suppose each parent carries both the G and Y genes M ={Male contributes G}; F ={Female contributes G} Are M and F independent?

16. Several Independent Events Events A 1, A 2, …, A n are mutually independent if for every k (k=2, 3, …, n) and every index set i 1, …, i k That is, events are mutually independent if the probability of the intersection of any subset of the n events is equal to the product of the individual probabilities