# Dependent Event Vs Independent Event Vs Mutually Exclusive Events

#### sachinner

##### New Member
While reading probability book I come across with above terms.
Following questions raised in my mind.

1)Are Dependent events always Mutually Exclusive and vice versa.
2)Are Independent events always Mutually Exclusive and vice versa.

Thanks!

#### hlsmith

##### Omega Contributor
Or no diagram. Does something sound wrong about this statement, "1)Are Dependent events always Mutually Exclusive and vice versa"? Ignoring the vice versa part.

#### BGM

##### TS Contributor
Two events $$A, B$$ are said to be independent (under probability $$P$$) if $$P(A \cap B) = P(A)P(B)$$.
Otherwise they are dependent.

Two events $$A, B$$ are said to be mutually exclusive if $$A \cap B = \varnothing$$.

Observe that any pairs of event with non-empty intersection (i.e. they are not mutually exclusive) can be either dependent or independent. The probability defined can be arbitrary and you can construct as many counter examples as you want.

For a pair of mutually exclusive events $$A, B$$, it is easy to see that

$$P(A \cap B) = P(\varnothing) = 0$$

Therefore in this case they are independent, i.e

$$P(A \cap B) = 0 = P(A)P(B)$$

if and only if at least one of the $$P(A), P(B)$$ equals to zero.

So for most of the non-trivial case, mutually exclusive means they are dependent.

#### sachinner

##### New Member
Thanks for response.
Vice versa included at the end to say that to confirm whether statements have two way definition.
i.e. are dependent events always mutually exclusive AND are mutually exclusive events always dependent.

The explanation seems quite theoretical to understand.
If you could explain with examples then it would be nice for anybody to understand easily.