Mutually exclusive events are those events that are events that cannot occur simultaneously at the same time. However, there are some events in statistics that cannot occur simultaneously anyway. For example, if you toss a coin then the two outcomes head and tail will not occur same time simultaneously. However, in mutually exclusive events, the outcomes that will come out is known to all but which outcome will come at first that is not known. This is because all the probability of the outcome is equal and they have equal chance of getting selected. In other words the mutually exclusive events are said to be disjoint events.

For example if A and B are two events then the mutually exclusive events can be claimed as P (A and B) =0. On the other hand, the rule of addition is only valid in probability if the events A and B are mutually exclusive events. If A and B are said to be mutually exclusive events then the probability of an event A occurring or the probability of event B occurring is given as P (A) + P (B) then  P (A or B) = P(A) + P(B). For example when a six sided dice is rolled out then the event of occurring 2 or 5 is mutually exclusive events as both the events cannot occur at the same time. It will give either 2 or 5 but both cannot be shown in one dice.

Mutually Exclusive

In order to prove the mutually exclusive events, the events A and B the simultaneous occurrence of both the events is not possible. P (AUB) =P (A) +P (B)-P (AՈB). This is an important relationship because if the events are to be mutually exclusive P (AՈB) needs to be 0 so that P(AUB)= P(A)+P(B). Now for example the lets us consider following set of events

S={1,2,3,4,5,6}, A={1,3,5} and B={2,4,6}. Now P (A) = 3/6 and P (B) = 3/6 and thus P (AUB) =1 thus P (AUB) = 3/6+3/6-0 thus LHS =RHS which claims that if mutually exclusive events to occur the common portion of the individual sets with the universal set should be zero. Now in order to increase the probability value of mutually exclusive events to occur it is highly important for the common part to be zero. Now through the selection of better sets of occurrence, there will be chance for the mutually exclusive events. Now in any case P(AUB)≠ P(A)+P(B) then the events cannot be claimed as mutually exclusive events. It becomes highly problematic to judge whether the events are mutually exclusive or not.