Question

31. Assume that we have two events, A and B. that are mutually exclusive. Assume further that we know P(A) 30 and P(B) a. What is P(A n B)? b. What is P(A I B)? c. 40. A student in statistics argues that the concepts of mutually exclusive events and inde- pendent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer. ake about mutually exclusive and independent events given the results of this problem?

Answer 1

Answer 31 -        P(A) = .30 , P(B) = .40 and A and B are mutually exclusive.

A – Value of P(A ꓵ B) = 0 because A and B are mutually exclusive thus they can’t occur together.

B – Value of P(A | B) = P(A or B)/P(B) = 0 because A and B are mutually exclusive thus they can’t occur together.

C – Definition of the mutual exclusivity defines that the events can’t occur together. P(A ꓵ B) = 0 the two events are independent when the occurrence of firs does not change the probability value of the occurrence of other P(A/B) = P(B). These values and concepts are different. To be mutually exclusive events the conditional probability of an event and the joint probability of A and B should equal to 0 in value.

D – If the events are mutually exclusive then the events are independent, but if the events are independent they may be or may not be mutually exclusive.

Answer – 39 = P(A1 ) = .40 , P(A2) = .60 , P(A1 ꓵ A2) = 0

A – Yes, they are mutually exclusive events because P(A1 ꓵ A2) = 0 and they are not occurring together.

B – P(B|A1) = P(A1 ꓵ B)/P(A1) = .20

P(A1 ꓵ B)/.40 = .20

         P(A1 ꓵ B) = .20 * .40

        P(A1 ꓵ B) = .08

P(B|A2) = P(A2 ꓵ B)/P(A2) = .05

               P(A2 ꓵ B)/.60 = .05

               P(A2 ꓵ B) = .05 * .60

               P(A2 ꓵ B) = .03

C - P(B) = P(P(A1 ꓵ B) + P(A2 ꓵ B)

P(B) = .08+.03

P(B) = .83

D – Bayes’ theorem to compute P(A1 | B) = P(A1) P(B|A1)/P(B)

= (.40*.08)/.83

   = 0.0385

Bayes’ theorem to compute P(A2 | B) = P(A2) P(B|A2)/P(B)

                     = (.60 * .03)/.83

                      = .0216