# Independent Events

If the occurrence of one event has no effect on the probability of another event, the two events are independent events .

Example :

Consider the experiment of tossing a coin twice.  The sample space for the experiment is $\left\{\left(\text{H,H}\right),\left(\text{H,T}\right),\left(\text{T,H}\right),\left(\text{T,T}\right)\right\}$ .  Let $A$ be the event that a tail is obtained on the first toss and $B$ be the event that a head is obtained on the second toss.

$A=\left\{\left(\text{T,H}\right),\left(\text{T,T}\right)\right\}$ and $B=\left\{\left(\text{H,H}\right),\left(\text{T,H}\right)\right\}$

$A$ and $B$ are not mutually exclusive because $A\cap B=\left\{\left(\text{T,H}\right)\right\}$ but they are independent because obtaining a tail on the first toss does not affect the outcome of the second toss.

$\begin{array}{l}P\left(A\right)=\frac{2}{4}=\frac{1}{2}\\ P\left(B\right)=\frac{2}{4}=\frac{1}{2}\\ P\left(A\cap B\right)=P\left(A\right)\cdot P\left(B\right)=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}\end{array}$

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