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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:

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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.

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

(iii) The event A∩B 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

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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.

I will be needing some assistance in this topic so I could understand what it is all about.

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