Suppose A and B are events with 0 LT PA LT 1 and 0 LT PB LT

Suppose A and B are events with 0 LT P(A) LT 1 and 0 LT P(B) LT 1. If A and B are disjoint, can they be independent? If A and B are independent, can they be disjoint? If A B, can A and B be independent? If A and B are independent, can A and A B be independent?

Solution

Note that 1> P(A), P(B) >0

Two events A and B independent if and only if P(both A and B)=P(A)*P(B)

(a)

P(A and B)=0 (as A, B are disjoint)

But, P(A)*P(B) > 0

So, A and B are not independent.

(b)

A and B are independent.

P(A and B)=P(A)*P(B) > 0

So, A and B are not disjoint as P(A and B)>0

(c)

P(A and B)=P(A)

because A is a subset of B and thus, lies inside B. Whenever, A occurs B also occurs. That is, every element (or every outcome) of A also belongs to the event [A and B]. And obviously, every element of [A and B] is in A. So, the event [both A and B]=[B]

Now, P(A)> P(A)*P(B) (as P(B)<1)

So, P(A and B)=P(A)>P(A)*P(B)

Hence, A and B are not independent.

(d)

A is a subset of C= [A or B]

From part (c) A and C are not independent. That is, A and [A or B] are not independent.