Mutually exclusive events and independent events are two concepts commonly used in probability. From how they sound, these two concepts may appear similar to many people – excluding and independent. Here, we explore, from first principles, what they are and how they are related.
The definitions
Two events are mutually exclusive, if one happens, the other can not occur. Or, if the probability of their intersection is zero. If A and B are the two events,
P(A∩B) = 0
Turning left and turning right are mutually exclusive.
While flipping a coin, the occurrence of the head and the occurrence of the tail are mutually exclusive events.
If two events are independent, the probability of their intersection is equal to the product of the two probabilities.
P(A∩B) = P(A)P(B)
There is another definition which may be more intuitive to some. The occurrence of the other does not influence the probability of one event. The probability of A given B equals the probability of A.
P(A|B) = P(A)
The relationship
Consider two mutually exclusive events, which implies P(A∩B) = 0. They are independent, i.e., P(A)P(B) = 0, if only if P(A) or P(B) or both = 0. Suppose both probabilities are > 0, then P(A)P(B) > 0. They are not independent.
Therefore, if A and B are mutually exclusive with positive probabilities, then they are not independent.
Reference
Are mutually exclusive events independent?: jbstatistics
