I need help in solving this probability question. Please explain the solutions to these questions.

Question 1

a) Suppose a sample space contains 6 outcomes denoted by e1, e2, . . . , e6. Let A, B and C be the events A = {e1, e2, e3}, B = {e3, e4, e5}, and C = {e5, e6}, and suppose probabilities are assigned as follows:

P(A) = 0.4 , P(B) = 0.5 , P(C) = 0.5 , P(A B) = 0.2 , P(B C) = 0.2 .

(i) By using a Venn diagram, or otherwise, verify that such an assignment of probabilities is possible.

(ii) Find the probabilities of the events A B, A B C and C B.

(iii) The event AB can be described in words as "both the event A and the event B occur". Describe similarly in words, the three events of part (ii).

b) (i) What is wrong with the following assignment of probabilities?

P (A) = 0.6, P (B) = 0.3, P (A B) = 0.4.

(ii) What is wrong with the following assignment of probabilities?

P (A) = 0.6, P (B) = 0.7, P (A B) = 0.5.

c) Of three events A, B, and C, suppose A and B are independent and B and C are disjoint. Their probabilities are P(A) = 0.6, P(B) = 0.2 and P(C) = 0.2. Calculate the probabilities of the following events:

(i) Both B and C occur.

(ii) At least one of A and B occurs

(iii) B does not occur.

(iv) All three events occur.