More About Probability 5.3 More About Probability

Remember the 5 Rules 0 < P(A) < 1 for any event A P(S) = 1 Complement Rule P(A ) = 1 – P(A) Addition Rule for Disjoint A & B, P (A or B) = P(A) + P(B) 5. Multiplication Rule for Independent A & B, P (A and B) = P(A) P(B) c

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

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

General Addition Rule For Unions of Two Events Suppose that events A and B are not disjoint, they can occur simultaneously P(A or B) = P(A) + P(B) – P(A and B) The Probability of the union is less than the sum of their probabilities if they are not disjoint

If A and B are disjoint, the event {A and B} that both occur has no outcomes in it. This ‘empty event’ is the complement of the sample space S and must have probability 0.

A B A and B P(A or B) = P(A) + P(B) – P(A and B) for any events A and B A B A and B

Ex. ) Matt and Sarah are up for promotion at a law firm Ex.) Matt and Sarah are up for promotion at a law firm. Matt guesses that his probability of promotion = .7 and Sarah has a chance of = .5. Also, he guesses that the probability that both him and Sarah become partners = .3 P(A) + P(B) – P(A and B) P(at least one is promoted) = .7 + .5 - .3 = .9 the P(neither is promoted) = .1 by Complement Rule

What is the probability that Matt gets promoted but not Sarah? Would be prob. of Matt promoted minus prob. both promoted by subtracting areas in Venn Diagram

The simultaneous occurrence of two events such as A= Matt is promoted and B= Sarah is promoted is called a Joint Event.

The probability of a Joint Event such as P(Matt promoted and Sarah is promoted) = P(A and B) is called Joint Probability.