Independence

Definition Two events A and B are independent if and are dependent otherwise. The definition implies that B is independent of A as well, for if A is independent of B,

Example Consider a gas station with six pumps numbered 1,2,…,6, and let denote the simple event that a randomly selected customer uses pump j, j=1, …,6. Suppose that , , and . Let , , We can show that A and B are dependent, but A and C are independent.

Multiplication rule for probability of an intersection A and B are independent if and only if (iff) The last equation follows from the general multiplication rule for the probability of the intersection of two events and the definition of independence that we gave previously.

Independence of more than two events Events are mutually independent if for every k and every subset of the indices ,

Example An aircraft seam requires 25 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability. (a) If 20% of all seams need reworking, what is the probability that a rivet is defective? How small should the probability of a defective rivet be to ensure that only 10% of all seams need reworking?

Solution (take 1-p) (take is 1-p) Here p is the probability that a rivet works.