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Statistics 400 - Lecture 4 Today - 4.1-4.5 Suggested Problems: 2.1, 2.48 (also compute mean), construct histogram of data in 2.48 Probability (Chapter 4) “There is a 75% chance of rain tomorrow” What does this mean? Definitions Probability of an outcome is a numerical measure of the chance of the outcome occurring A experiment is random if its outcome is uncertain Sample space, S, is the collection of possible outcomes of an experiment Event is a set of outcomes Event occurs when one of its outcomes occurs Example A coin is tossed 2 times S= Describe event of getting 1 heads and 1 tails Probability of an event is the long-term proportion of times the event would occur if the experiment is repeated many times Probability of event, A is denoted P(A) 0 P( A) 1 P(A) is the sum of the probabilities for each outcomes in A P(S)=1 Discrete Uniform Distribution Sample space has k possible outcomes S={e1,e2,…,ek} Each outcome is equally likely P(ei)= If A is a collection of distinct outcomes from S, P(A)= Bag of balls has 5 red and 5 green balls 3 are drawn at random S= A is the event that at least 2 green are chosen A= P(A)= Example (pg 140) Inherited characteristics are transmitted from one generation to the next by genes Genes occur in pairs and offspring receive one from each parent Experiment was conducted to verify this idea Pure red flower crossed with a pure white flower gives Two of these hybrids are crossed. Outcomes: Probability of each outcome Sometimes, not all outcomes are equally likely (e.g., fixed die) Recall, probability of an event is long-term proportion of times the event occurs when the experiment is performed repeatedly NOTE: Probability refers to experiments or processes, not individuals Probability Rules Have looked at computing probability for events How to compute probability for multiple events? Example: 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? Addition Rule: P( A B) P( A) P( B) P( A B) If two events are mutually exclusive: P( A B) P( A) P( B) Complement Rule P( A) 1 P( A ) 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: P( A B) P( A | B) Called “probability of A given B ” P( B) 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? Multiplication rule for conditional probability: P( A B) P( A | B) P( B) Can use any 2 of the probabilities to get the third Independent Events Two events are independent if: P( A | B) P( A) The intuitive meaning is that the outcome of event B does not impact the probability of any outcome of event A Alternate form: P( A and B) P( A) P( B) Example Flip a coin two times S= A={head observed on first toss} B={head observed on second toss} Are A and B independent? 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? 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