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 25-64 65+ total Married 3,046 48,116 7,767 58,929 Never Married 9,289 9,252 768 19,309 Widowed 19 2,425 8,636 11,080 Divorced 260 8,916 1,091 10,267 Total 12,614 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!)

B High School Athlete A B B A B Tree Diagram Professional College c 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 P(B and A ) = P(A ) P(B A ) = . c c c

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)