Independent and Mutually Exclusive Events

Date: 01/03/2007 at 11:10:41
From: TR
Subject: What is  difference between independent and exclusive events

I am confused about independent and mutually exclusive events.  Can
you please provide a practical example of each?

If A and B are independent then it means that the occurrence of one 
event has no effect on the occurrence of the other event.  I think 
mutually exclusive also means the same.  Is that right?

Date: 01/03/2007 at 15:43:39
From: Doctor Pete
Subject: Re: What is  difference between independent and exclusive events

Hi TR,

If two events A and B are independent, then

  Pr[A and B] = Pr[A]Pr[B];

that is, the probability that both A and B occur is equal to the
probability that A occurs times the probability that B occurs.

If A and B are mutually exclusive, then

  Pr[A and B] = 0;

that is, the probability that both A and B occur is zero.  Clearly, if 
A and B are nontrivial events (Pr[A] and Pr[B] are nonzero), then they 
cannot be both independent and mutually exclusive.

A real-life example is the following.  Consider a fair coin and a fair 
six-sided die.  Let event A be obtaining heads, and event B be rolling 
a 6.  Then we can reasonably assume that events A and B are 
independent, because the outcome of one does not affect the outcome of 
the other.  The probability that both A and B occur is

  Pr[A and B] = Pr[A]Pr[B] = (1/2)(1/6) = 1/12.

Since this value is not zero, then events A and B cannot be mutually
exclusive.

An example of a mutually exclusive event is the following:  Consider
a fair six-sided die as before, only in addition to the numbers 1 
through 6 on each face, we have the property that the even-numbered
faces are colored red, and the odd-numbered faces are colored green.
Let event A be rolling a green face, and event B be rolling a 6.  Then

  Pr[A] = 1/2
  Pr[B] = 1/6

as in our previous example.  But it is obvious that events A and B
cannot simultaneously occur, since rolling a 6 means the face is red,
and rolling a green face means the number showing is odd.  Therefore

  Pr[A and B] = 0.

Therefore, we see that a mutually exclusive pair of nontrivial events
are also necessarily dependent events.  This makes sense because if A
and B are mutually exclusive, then if A occurs, then B cannot also
occur; and vice versa.  This stands in contrast to saying the outcome
of A does not affect the outcome of B, which is independence of events.

- Doctor Pete, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/04/2007 at 09:42:13
From: TR
Subject: Thank you (What is  difference between independent and
exclusive events)

Thank you very much for your reply.  It was very helpful!