Mutually Exclusive And Independent Events

Mutually exclusive events:

The two events A & B are said to be mutually exclusive events when they cannot happen at the same time.

Thus, A & B has no intersection area since they are not occurring at the same time.

P(AΠB) = P(AB) = 0

where P(AΠB) is the probability of A intersection B and it is same as the probability of A multiplied by B, P(AB).

Independent events:

The two events A & B are said to be independent events when the occurrence of one event does not influence the occurrence of other event.

In this case, P(A Π B) = P(AB) = P(A)*P(B), that is, the probability of A & B is the probability of A multiplied by the probability of B.

Difference between mutually exclusive events and independent events:

One difference is as we have seen above, if A & B are mutually exclusive, the probability of their intersection is zero. P(AB) = 0 and 

if they are independent, P(AB) = P(A)*P(B).

The other difference can be seen from conditional probability as below:

P(A/B) = P(B/A) = 0 in case of mutually exclusive events and

P(A/B) = P(A) & P(B/A) = P(B) in case of independent events.

For mutually exclusive events, 

P(A/B) = P(AΠB)/P(B) = 0/P(B) = 0 and

P(B/A) = P(BΠA)/P(A) = 0/P(A) = 0.

Here, the logic is this --- A/B i.e, A given B means occurrence of A when B is occurring or existing already at the time of happening of event A which means both events happening together and this is opposite to mutually exclusive events as they cannot happen at the same time and thus P(A/B) is 0 for mutually exclusive events.

For independent events,

P(A/B) = P(AΠB)/P(B) = P(A)*P(B)/P(B) = P(A) and

P(B/A) = P(BΠA)/P(A) = P(B)*P(A)/P(A) = P(B).

Here the logic is this --- A given B means the occurence of A when B has been existing/occurring which suggests the influence of occurrence of B on the occurrence of A and independent events means no influence, the occurrence of event B does not influence the occurrence of event A and thus the probability of A given B remains same as the probability of event A itself. Similarly, P(B/A) is equal to the probability of event B itself as the occurrence of A doesn't influence the occurrence of B as they are independent events.