A lot of 10 components contains 3 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let She the event that the second component drawn is defective.

a. Find P (A).

b. Find P (B|A).

c. Find P (A ∩ B).

d. Find P (Ac∩B).

e. Find P (B).

f. Are A and B independent? Explain.

There are 10 components with 3 that are defective:

A = event that the first component is defective

B = event that the second component is defective

b) P (A) = this is the probability that the first component is defective:

= number of defective component/total number of component

P (A) = 3/10 = 0.3

b) P (B|A) = the probability that the second component is defective after the first defective component. Since A has occurred:

= number of defective component left/total number of component left

P (B|A) = 2/9 = 0.22

c) P (A ∩ B) = probability that both are defective i. e. A and B occurring

= 3/10 * 2/9

P (A ∩ B) = 6/90 = 0.06

d) P (Ac ∩ B) = probability that A is not defective and B is defective

P (Ac) = 1 - 3/10 = 7/10 = 0.7

∴ P (Ac ∩ B) = 7/10 * 2/9 = 14/90 = 0.16

e) P (B) = probability that the second component is defective

= 2/9 = 0.22

f) B is dependent on A because the defective component is not replaced after the first selection. There would be 9 items left after the first selection of which 2 are defective.