Statistics General Probability Rules

Union The union of any collection of events is the event that at least one of the collection occurs The union of any collection of events is the event that at least one of the collection occurs

Addition Rule for Disjoint Events If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then P(one or more of A, B, C) = P(A) + P(B) + P(C) P(one or more of A, B, C) = P(A) + P(B) + P(C)

Not Disjoint If events A and B are not disjoint, they can occur simultaneously. If events A and B are not disjoint, they can occur simultaneously. P(A or B) = P(A) + P(B) – P(A and B) P(A or B) = P(A) + P(B) – P(A and B)

Deborah and Marshall problem P 362, example 6.17 P 362, example 6.17 Deborah = 0.7 Deborah = 0.7 Matthew = 0.5 Matthew = 0.5 Both = 0.3 Both = 0.3 Question – What is the probability that at least one of them is promoted?

What is the probability of at least one of them is promoted? P(at least one) = – 0.3 P(at least one) = – 0.3 P(at least one) = 0.9 P(at least one) = 0.9

Table of probabilities Promoted Not promoted total DebPromoted Total0.51.0

Questions P(D and M) = P(D and M) = P(D and not M) = P(D and not M) = P(Not D and M) = P(Not D and M) = P(Not D and not M) = P(Not D and not M) =

Answers P(D and M) = 0.3 P(D and M) = 0.3 P(D and not M) = 0.4 P(D and not M) = 0.4 P(Not D and M) = 0.2 P(Not D and M) = 0.2 P(Not D and not M) = 0.1 P(Not D and not M) = 0.1

Problems to do 46, 53 46, 53

Conditional Probability the probability of an event happening knowing that another event has happened. the probability of an event happening knowing that another event has happened. Written as P(AlB) the probability of B happening knowing that A has happened. Written as P(AlB) the probability of B happening knowing that A has happened.

total Married7,84243,8088,27059,920 Never Married 13,9307, ,865 Widowed362,5238,38510,944 Divorced7049,1741,26311,141 Total22,51262,68918,669103,870

A = the woman chosen is young, ages 18 to 29 A = the woman chosen is young, ages 18 to 29 B = the woman chosen is married B = the woman chosen is married

P(A) = 22,512/103,870 = P(A) = 22,512/103,870 = P(A and B) = 7,842/103,870 = P(A and B) = 7,842/103,870 = 0.075

Probability she is married given that she is young. Probability she is married given that she is young. P(B l A) = 7,842/22,512 = P(B l A) = 7,842/22,512 = 0.348

General multiplication rule for two events P(A and B) = P(A)P(B l A) P(A and B) = P(A)P(B l A)

Definition of Conditional Probability

Problems 56, 58 56, 58

Extended Multiplication rules Intersection: the intersection of any collection of events is the event that all of the events occur. Intersection: the intersection of any collection of events is the event that all of the events occur.

Example The intersection of three events A, B, and C has the probability The intersection of three events A, B, and C has the probability P(A and B and C) = P(A and B and C) = = P(A)P(B|A)P(C|A and B) = P(A)P(B|A)P(C|A and B)

Future of High School Athletes Only 5% of male high school basketball, baseball and football players go on to play at the college level. Of these, only 1.7% enter major league professional sports. About 40% of the athletes who compete in college and then reach the pros have a career of more than 3 years. Only 5% of male high school basketball, baseball and football players go on to play at the college level. Of these, only 1.7% enter major league professional sports. About 40% of the athletes who compete in college and then reach the pros have a career of more than 3 years.

Events A = {competes in college} A = {competes in college} B = {competes professionally} B = {competes professionally} C = {pro career longer than 3 years} C = {pro career longer than 3 years} P(A) = 0.05 P(A) = 0.05 P(B|A) = P(B|A) = P(C|A and B) = 0.4 P(C|A and B) = 0.4

P(A and B and C) = P(A and B and C) = = P(A)P(B|A)P(C|A and B) = P(A)P(B|A)P(C|A and B) = 0.05 x x 0.40 = 0.05 x x 0.40 = = Only 3 out of every 10,000 high school athletes can expect to compete in college and have a career greater than 3 years Only 3 out of every 10,000 high school athletes can expect to compete in college and have a career greater than 3 years

Tree Diagrams The probability P(B) is the sum of the probabilities of the two branches ending at B.

Probability of reaching B given college is Probability of reaching B given college is 0.05x0.017= x0.017= Probability of reaching B not going to college is Probability of reaching B not going to college is 0.95x0.0001= x0.0001=

Probability of P(B) = Probability of P(B) = = = Or about 9 students out of 10,000 will play professional sports. Or about 9 students out of 10,000 will play professional sports.

Problems 64, 67, 70, 77, 79, 80, 83, 87