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Mutually Exclusive and Non-mutually Exclusive Events

Two events are called mutually exclusive if they can not happen at the same time. In other words, two mutually exclusive events do not intersect. For example, when we toss a coin, the possible outcomes are a head or a tail. Since these outcomes can not happen at the same time, they are mutually exclusive. 


Example: In an experiment, we roll a fair die. Event A is rolling an even number and event B is rolling an odd number. Are these events mutually exclusive? Represent these events on a Venn diagram.
The sample space is: . The two events A and B are and Since these events do not intersect, they are mutually exclusive. The Venn diagram for this problem is:




Two mutually exclusive events A and B satisfy:
Two events A and B are called non-mutually exclusive if their intersection is not zero. In other words, two non-mutually exclusive events can happen at the same time.
Example: In an experiment we roll a fair die. Suppose that event A is rolling an even number and event B is rolling a multiple of 3. Determine whether A and B are mutually exclusive.
Events A and B are:  We have:
Since A and B intersect, they are non-mutually exclusive. Here is the Venn diagram:




Two non-mutually exclusive events satisfy:
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