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5.3 Continued Conditional Probability • Similar to a conditional distribution for Chapter 4, that was the distribution of a variable given that a condition is satisfied. AGE 18-24 Married 3,046 Never Married 9,289 Widowed 19 Divorced 260 Total 12,614 25-64 65+ total 48,116 7,767 58,929 9,252 768 19,309 2,425 8,636 11,080 8,916 1,091 10,267 68,709 18,262 99,585 P(married) = = Would selecting a certain age range change the probability? 1. P(married 18-24) = * This is a Conditional Probability The probability of one event (the woman chosen is married) under the condition of another event (she is between ages 18-24) Read as “given the information that” 2. P(age 18-24 and married) = 3. P(age 18-24) = • There is a relationship among these three probabilities. • The joint probability that a woman is both age 18-24 and married is the product of the probabilities that she is age 18-24 and that she is married given that she is age 18-24. • P(age 18-24 and married) = P(age 18-24) x P(married age 18-24) = From the chart = <as before> General Multiplication Rule • The Joint Probability that both of two events A and B happen together can be found by P(A and B) = P(A) P(B A) P(B A) is the conditional probability that B occurs given the info that A occurs • With algebra we get Definition of Conditional Probability When P(A) > 0 , the conditional probability of B given A is P(B A) = P(A and B) P(A) • Ex. 1) What is the conditional probability that a woman is a widow, given that she is at least 65 years old? P(at least 65) = P(widowed and at least 65) = = = The conditional probability P(widowed at least 65) = P(widowed and at least 65) P(at least 65) = Check results from actual table = = • The Intersection of any collection of events is the event that all the events occur • The Multiplication Rule extends to the probability that all of several events occur. P(A and B and C) = P(A) x P(B A) x P(C A and B) • Ex.) 5% of male high school basketball/ baseball/ football players go on to play in college • Of these, 1.7 % enter majors. About 40% of the athletes who compete in college and reach the pros have a career of 3+ years. A = {compete in college} B = {compete in professional} C = {pro. Career 3+ years} • What is the probability that a high school athlete competes in college and then goes on to have a professional career in 3+ years? P(A) = P(B A) = P(C|A and B) = P(A and B and C) = . (3 out of 10,000!) Tree Diagram Professional College A B B High School Athlete c B A c B c • There are 2 disjoint paths to B (play professionally) • By the addition rule, P(B) = sum of those probabilities • Probability of reaching B through college is P(B and A) = P(A) P(B A) = • Probability of reaching B without college is c c c P(B and A ) = P(A ) P(B A ) = . • Final result P(B) = .00085 + .000095 = .000945 = 9/10,000 High school athletes play professionally Tree diagrams combine the addition and multiplication rules. • Mult. Rule states: Probability reaching the end of any branch is the product of the probabilities written on segments. • The product of any outcome, such as the event B, an athlete reaches professional sports is then found by adding probabilities of all branches that are a part of that event. Two events A & B, both pos. probability and independent if, P(B A) = P(B)