3.6 The Concept of Independence

Intuitively, if event B has no effect on event A we can say that A is independent of B. More precisely, if knowledge that event B has occurred does not affect the probability of A, we say that A is independent of B. This is expressed mathematically as:

Replacing P( A B) by its expression from Definition 3.4.1, we find:

This can be rewritten P( A ? B) = P( A) P( B).

It is interesting to note that the last expression implies that

In other words, if A is independent of B, then B is independent of A. Thus, we can say that if P( A ? B) = P( A) P( B) then A and B are independent.

This definition can be extended to more than two events. We say that A ₁, A ₂, , A _n are independent if

Example

1. Returning to the example in Section 3.4, we can ask whether the event window open is independent of the event door open . We have

   
   
   
   
   
   

   But

   
   
   
   
   
   

   Thus, we see that in the situation considered knowing that the door is open decreases the probability that the window will be open, and the two events are not independent.

2. Suppose that we know that the probability that a smoke...