Question

1. Consider events A, B, and C, with P(A) > P(B) > P(C) > 0. Events A and B are mutually exclusive and collectively exhaustive. Events A and C are independent.

(a) Can events C and B be mutually exclusive? Explain your reasoning. (Hint: You might find it helpful to draw a Venn diagram.)

(b)  Are events B and C independent? Explain your reasoning.

Answer 1

A and B are mutually exclusive and exhaustive , so :

P[A]+P[B] = 1

a) C and B can be mutually exclusive , if C is subset of A .

b)

If (a) is true , that is C is subset of A

B and C will be independent if :

Now ,   as , A and B are mutually exclusive and C is subset of A , so C and B are also mutually exclusive .

But P[B] >0 and P[C] > 0  

so,  

so, B and C can't be independent .

if (a) is not true , that is C is subset of B

Then obviously

so, B and C cannot be independent .