Mutually Exclusive

A Venn diagram depicting two mutually exclusive events.
Events A and B are Mutually Exclusive. C.K.Taylor

In probability two events are said to be mutually exclusive if and only if the events have no shared outcomes. If we consider the events as sets, then we would say that two events are mutually exclusive when their intersection is the empty set. We could denote that events A and B are mutually exclusive by the formula AB = Ø. As with many concepts from probability, some examples will help to make sense of this definition.

Rolling Dice

Suppose that we roll two six-sided dice and add the number of dots showing on top of the dice. The event consisting of "the sum is even" is mutually exclusive from the event "the sum is odd."  The reason for this is because there is no way possible for a number to be even and odd.

Now we will conduct the same probability experiment of rolling two dice and adding the numbers shown together. This time we will consider the event consisting of having an odd sum, and the event consisting of having a sum greater than nine. These two events are not mutually exclusive.

The reason why is evident when we examine the outcomes of the events. The first event has outcomes of 3, 5, 7, 9 and 11. The second event has outcomes of 10, 11 and 12. Since 11 is in both of these, the events are not mutually exclusive.

Drawing Cards

We illustrate further with another example. Suppose we draw a card from a standard deck of 52 cards.

Drawing a heart is not mutually exclusive to the event of drawing a king. This is because there is a card (the king of hearts) that shows up in both of these events.

Why Does It Matter

There are times when it is very important to determine if two events are mutually exclusive or not. Knowing whether two events are mutually exclusive influences the calculation of the probability that one or the other occurs.

Go back to the card example. If we draw one card from a standard 52 card deck, what is the probability that we have drawn a heart or a king?

First break this into the individual events. To find the probability that we have drawn a heart, we first count the number of hearts in the deck as 13, and then divide by the total number of cards. This means that the probability of a heart is 13/52.

To find the probability that we have drawn a king we start by counting the total number of kings, resulting in as four, and next divide by the total number of cards, which is 52. The probability that we have drawn a king is 4/52.

The problem is now to find the probability of drawing either a king or a heart. Here’s where we must be careful. It is very tempting to simply add the probabilities of 13/52 and 4/52 together. This would not be correct because the two events are not mutually exclusive. The king of hearts has been counted twice in these probabilities. To counteract the double counting, we must subtract the probability of drawing a king and a heart, which is 1/52. Therefore the probability that we have drawn either a king or a heart is 16/52.

Other Uses of Mutually Exclusive

A formula known as the addition rule gives an alternate way to solve a problem such as the one above.

The addition rule actually refers to a couple of formulas that are closely related to one another. We must know if our events are mutually exclusive in order to know which addition formula is appropriate to use.