A die marked $1,2,3$ in red and $4,5,6$ in green is tossed. Let $A$ be the event, ‘the number is even,’ and $\mathrm{B}$ be the event, ‘the number is red’. Are A and B independent?

Solution:

The dice sample space will be be $S={1,2,3,4,5,6}$

Let $A$ be the event, the number is even, sample space of the event:

$\Rightarrow \mathrm{A}={2,4,6}$

$\Rightarrow P(A)=3 / 6=1 / 2$

Now, Let $\mathrm{B}$ be the event, the number is red, sample space of event B,

$\Rightarrow B={1,2,3}$

$\Rightarrow P(B)=3 / 6=1 / 2$

As, A $\cap B={2}$

$\Rightarrow P(A \cap B)=1 / 6 \ldots \ldots . .$

And $P$ (A). $P(B)=1 / 2 \times 1 / 2=1 / 4 \ldots . .$ (2)

From (1) and (2) $P(A \cap B) \neq P(A) . P(B)$

Therefore, $A$ and $B$ are not independent events.