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**Unformatted text preview: **Conditional Probability Multiplication Rule Law of Total Probability. Bayes Rule. Independence. Random Variables. Title Page JJ II J I Page 1 of 27 Go Back Full Screen Close Quit ORIE 270 Engineering Probability and Statistics Lecture 5: Conditional Probability Sidney Resnick School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA http://legacy.orie.cornell.edu/ ∼ sid [email protected] September 5, 2007 Conditional Probability Multiplication Rule Law of Total Probability. Bayes Rule. Independence. Random Variables. Title Page JJ II J I Page 2 of 27 Go Back Full Screen Close Quit 1. Conditional Probability Suppose we have a probability model ( S, A , P ) reflecting our estimates of likelihood of outcomes and then we get additional information that some event must occur. What to do? Examples: • Example 1: Suppose 270 consists of – W women, M males; – WS female soccer players – MS male soccer players. Each member of the population is characterized by 2 traits: (gender, play soccer or not). Experiment: select a student from this population. Probability model for this experiment (w=female, m=male, s=soccer player, n=not) S (w,s) (w,ns) (m,s) (m,ns) P WS W + M W- WS W + M MS W + M M- MS W + M Conditional Probability Multiplication Rule Law of Total Probability. Bayes Rule. Independence. Random Variables. Title Page JJ II J I Page 3 of 27 Go Back Full Screen Close Quit Now restrict attention to the subpopulation of women. Within this subpopulation, we only distinguish by one trait: play soccer or not. The model changes to S (w,s) (w,ns) P WS W W- WS W OR S (w,s) (w,ns) (m,s) (m,ns) P WS W + M / W M + W W- WS W + M / W M + W Note → In the first table, we calculate WS W = % of women students who are soccer players by restricting attention to the subpopulation of women but as we see in the 2nd table, it is also possible to do the calculation relative to the orignal population of both males and females: Conditional Probability Multiplication Rule Law of Total Probability. Bayes Rule. Independence. Random Variables. Title Page JJ II J I Page 4 of 27 Go Back Full Screen Close Quit → The ratio WS W + M W W + M = % of class consisting of female soccer players % of class consisting of females effectively does the calculation relative to the bigger popula- tion of the whole class but renormalizes the percentages. • Example 2: In the skunky beer example, suppose I am second in line and I get additional information that the first in line received a good (or bad) beer. How do I revise my probability estimates? Definition: Given the model ( S, A , P ), the conditional probability of A given B is P ( A | B ) = P ( AB ) P ( B ) provided P ( B ) > 0. [ Reminder: AB = A ∩ B .] Could regard the new sample space as B but usually don’t and simply redistribute the probability onto B , making sure B gets 0 probability....

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