# Mutually exclusive events are also independent??

Mutually exclusive events are also independent or not ? can some one explain with an example? Is it compulsory for independent event to be mutually exclusive? what are the relation between both ?

No, actually if two events are mutually exclusive it is only in a special case that they can be independent. The definitions state that two events $A,B$ are

• mutually exclusive iff: $P(A\cap B)=0$, i.e. $A\cap B =\emptyset$,
• independent iff: $P(A\cap B)=P(A)P(B)$.

Assume know that $A,B$ are both mutually exclusive and independent. Then combining the above yields $$P(A)P(B)=0$$ which implies that at least one of the $P(A)$ and $P(B)$ has to be zero.

Informal examples: The events $A=$ I will roll a $6$ and $B=$ I will roll a $2$ are mutually exclusive. They cannot occur simultaneously. Similarly the events "rain" and "no rain" etc.

On the other hand the events "rain" and "I will roll a $6$" can occur simultaneously (i.e. the occurrence of the one does not exclude the occurrence of the other) but they are obviously independent, i.e. if you are given the information that it is "raining" or not you cannot in anyway infer whether I will roll a $6$ or not and vice versa.

Mutually exclusive and independent are almost opposites of each other. If they are mutually exclusive then if one happens the other cannot happen - quite the opposite of being independent. Independence essentially means that if one event happens it has no effect on whether the other event happens. Mutual exclusivity normally arises when considering the outcomes of ONE experiment e.g roll one die. Then "roll is less than 2" and "roll is greater than 4" are mutually exclusive. Independence normally arises when considering the outcomes of more than 1 experiment, e.g roll 2 dice then "roll on die 1 is less than 2" and "roll on die 2 is greater than 4" are independent.