What is the difference between independent and mutually exclusive events?

Question

Two events are mutually exclusive if they can't both happen.

Independent events are events where knowledge of the probability of one doesn't change the probability of the other.

Are these definitions correct? If possible, please give more than one example and counterexample.

Answer 1

Yes, that's fine.

Events are mutually exclusive if the occurrence of one event excludes the occurrence of the other(s). Mutually exclusive events cannot happen at the same time. For example: when tossing a coin, the result can either be heads or tails but cannot be both.

$$\left.\begin{align}P(A\cap B) &= 0 \\ P(A\cup B) &= P(A)+P(B)\\ P(A\mid B)&=0 \\ P(A\mid \neg B) &= \frac{P(A)}{1-P(B)}\end{align}\right\}\text{ mutually exclusive }A,B$$

Events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of the other(s). For example: when tossing two coins, the result of one flip does not affect the result of the other.

$$\left.\begin{align}P(A\cap B) &= P(A)P(B) \\ P(A\cup B) &= P(A)+P(B)-P(A)P(B)\\ P(A\mid B)&=P(A) \\ P(A\mid \neg B) &= P(A)\end{align}\right\}\text{ independent }A,B$$

This of course means mutually exclusive events are not independent, and independent events cannot be mutually exclusive. (Events of measure zero excepted.)

Answer 2

If I toss a coin twice, the result of the first toss and the second toss are independent.

However the event that you get two heads is mutually exclusive to the event that you get two tails.

Suppose two events have a non-zero chance of occurring.

Then if the two events are mutually exclusive, they can not be independent.

If two events are independent, they cannot be mutually exclusive.