Refer to question 1 above. If the die were fair, determine whether or not the events A and B are independent.

Referring to the above Solution

we have
$A = \{ (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)\}$

$\Rightarrow$ $n(A) = 6$ and $n(S) = {6^2} = 36$

[where, $S$ is sample space]
$\therefore$ $P(A) = \frac{{n(A)}}{{n(S)}} = \frac{6}{{36}} = \frac{1}{6}$

and $B = \{ (4,6),(6,4),(5,5),(6,5),(5,6),(6,6)\}$

$\Rightarrow$ $n(B) = 6$ and $n(S) = {6^2} = 36$
$\therefore$ $P(B) = \frac{{n(B)}}{{n(S)}} = \frac{6}{{36}} = \frac{1}{6}$

Also, $A \cap B = \{ (5,5),(6,6)\}$
$\Rightarrow$ $n(A \cap B) = 2$ and $n(S) = 36$

$\therefore$ $P(A \cap B) = \frac{2}{{36}} = \frac{1}{{18}}$

Also, $P(A) \cdot P(B) = \frac{1}{{36}}$

Thus $P(A \cap B) \ne P(A) \cdot P(B)$

So, we can say that both A and B are not independent events.