1. Statistics 400 - Lecture 4

2. zToday - 4.1-4.5 zSuggested Problems: 2.1, 2.48 (also compute mean), construct histogram of data in 2.48

3. Probability (Chapter 4) z“There is a 75% chance of rain tomorrow” zWhat does this mean?

4. Definitions zProbability of an outcome is a numerical measure of the chance of the outcome occurring zA experiment is random if its outcome is uncertain zSample space, S, is the collection of possible outcomes of an experiment zEvent is a set of outcomes zEvent occurs when one of its outcomes occurs

5. Example zA coin is tossed 2 times zS= zDescribe event of getting 1 heads and 1 tails

6. zProbability of an event is the long-term proportion of times the event would occur if the experiment is repeated many times zProbability of event, A is denoted P(A) z z P(A) is the sum of the probabilities for each outcomes in A zP(S)=1

7. Discrete Uniform Distribution zSample space has k possible outcomes S={e 1,e 2,…,e k } zEach outcome is equally likely zP(e i )= zIf A is a collection of distinct outcomes from S, P(A)=

8. zBag of balls has 5 red and 5 green balls z3 are drawn at random zS= zA is the event that at least 2 green are chosen zA= zP(A)=

9. Example (pg 140) zInherited characteristics are transmitted from one generation to the next by genes zGenes occur in pairs and offspring receive one from each parent zExperiment was conducted to verify this idea zPure red flower crossed with a pure white flower gives zTwo of these hybrids are crossed. Outcomes: zProbability of each outcome

10. zSometimes, not all outcomes are equally likely (e.g., fixed die) zRecall, probability of an event is long-term proportion of times the event occurs when the experiment is performed repeatedly zNOTE: Probability refers to experiments or processes, not individuals

11. Probability Rules zHave looked at computing probability for events zHow to compute probability for multiple events? zExample: 65% of Umich Business School Professors read the Wall Street Journal, 55% read the Ann Arbor News and 45% read both. A randomly selected Professor is asked what newspaper they read. What is the probability the Professor reads one of the 2 papers?

12. zAddition Rule: zIf two events are mutually exclusive: zComplement Rule

13. Conditional Probability zSometimes interested in in probability of an event, after information regarding another event has been observed zThe conditional probability of an event A, given that it is known B has occurred is: zCalled “probability of A given B ”

14. Example zIn a region 12% of adults are smokers, 0.8% are smokers with emphysema and 0.2% are non-smokers with emphysema zWhat is the probability that a randomly selected individual has emphysema? zGiven that the person is a smoker, what is the probability that the person has emphysema?

15. zMultiplication rule for conditional probability: zCan use any 2 of the probabilities to get the third

16. Independent Events zTwo events are independent if: zThe intuitive meaning is that the outcome of event B does not impact the probability of any outcome of event A zAlternate form:

17. Example zFlip a coin two times zS= zA={head observed on first toss} zB={head observed on second toss} zAre A and B independent?

18. Example zMendel used garden peas in experiments that showed inheritance occurs randomly zSeed color can be green or yellow z{G,G}=Green otherwise pea is yellow zSuppose each parent carries both the G and Y genes zM ={Male contributes G}; F ={Female contributes G} zAre M and F independent?

19. Example (Randomized Response Model) zCan design survey using conditional probability to help get honest answer for sensitive questions zWant to estimate the probability someone cheats on taxes zQuestionnaire: y1. Do you cheat on your taxes? y2. Is the second hand on the clock between 12 and 3? y YES NO zMethodology: Sit alone, flip a coin and if the outcome is heads answer question 1 otherwise answer question 2