# Is it true that non mutually exclusive events can only be independent events?

Aug 9, 2017

No. It is not always true. See explanation.

#### Explanation:

First we have to look at the definitions.

Two events $A$ and $B$ are independent if and only if $P \left(A \cap B\right) = P \left(A\right) \times P \left(B\right)$

Two events are mutualy exclusive if their product is empty set
(i.e. $P \left(A \cap B\right) = 0$)

Let's consider 2 events with throwing a 6 sided die.

$A$ - a result is not more than $5$ and

$B$ - a result is more than $4$

These 2 events are not mutualy exclusive because their product is:

$A \cap B$ - the result is more than $4$ and not more than $5$.

so: $A \cap B = \left\{5\right\}$, $P \left(A \cap B\right) = \frac{1}{6}$

Now let's check if the events are independent:

$A = \left\{1 , 2 , 3 , 4 , 5\right\}$, $B = \left\{5 , 6\right\}$, so:

$P \left(A\right) = \frac{5}{6}$, $P \left(B\right) = \frac{2}{6} = \frac{1}{3}$, so:

$P \left(A\right) \times P \left(B\right) = \frac{5}{6} \times \frac{1}{3} = \frac{5}{18}$

$P \left(A\right) \times P \left(B\right) \ne P \left(A \cap B\right)$, so the events are not independent.

Conclusion:

Example events are not mutualy exclusive and not independent, so it is a counter-example against hypothesis stated in the question.