Independent and Mutually Exclusive EventsDate: 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! |
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