# how to prove that mutually exclusive events are dependent events

I am trying to prove that 2 mutually exclusive events are always independent and that the opposite is not always true, meaning that if I have two independent events they are not necessary mutually exclusive.

For mutuall exclusive events P(A and B) = 0

For independent events P(A and B) = P(A)P(B)

I would like an example of two independent events that are not mutually exclusive

• Draw a card from a standard deck. Let $A$ be the event it's an ace and $B$ be the event the card is black. May 7, 2014 at 17:03

## 3 Answers

Two mutually exclusive events are neither necessarily independent nor dependent. For example, the events that a coin will come up head or that it will come up tail are exclusive, but not independent, because $P(H \text{ and } T) = 0$, whereas $P(H)P(T) = \frac{1}{4}$. On the other hand, any event $A$ is independent from the empty event $\emptyset$, because $P(\emptyset) = 0$, so $P(A \cap \emptyset) = P(\emptyset) = 0 = P(A)P(\emptyset)$, and $A$ is of course mutually exclusive from the empty event. For an example of two independent events that are not mutually exklusive, suppose you throw a coin two times. The event that the first coin comes up head is independent from the event that the second comes up head, because $$P(X_1 = H \text{ and } X_2 = H) = P(X_1 = H)P(X_2 = H),$$ but the events are not mutually exclusive, because of course $(X_1 = H \text{ and } X_2 = H)$ lies in both events.

Why you are trying to prove that on the first place. Mutually Exclusive(ME) and Independence are two different concepts. One doesn't implies the other and vice versa. In other words, two ME events (A and B) can be independent only if $P(A).P(B) = 0$.

An example of two events that are independent but not mutually exclusive are, 1) if your on time or late for work and 2) If its raining or not raining. These two events can occur at the same time (not mutually exclusive) however they do not affect one another.

• Welcome to Math.SE. On its face your proposed Answer seems to be a misstatement. Perhaps you should clarify, but first read the previous Answers. The body of the Question appears to contain a misstatement as well, and you can see how others have tried to address this issue. Apr 21, 2017 at 18:05
• I wouldn't say those events are independent at all :-) Apr 21, 2017 at 18:06
• Indeed the chance of my being late to work is often increased by it raining, but this may be a personal shortcoming! Apr 21, 2017 at 18:07