Two dice are tossed. Find whether the following two events A and B are independent $A = \{ (x,y):x + y = 11\}$ and $B = \{ (x,y):x \ne 5\}$,

where $(x,y)$ denotes a typical sample point.

We have, $A = \{ (x,y):x + y = 11\}$ and $B = \{ (x,y):x \ne 5\}$

$\therefore$ $A = \{ (5,6),(6,5)\} ,B = \{ (1,1),(1,2),(1,3),(1,4),(1,5)(1,6),$

$(2,1),(2,2),(2,3),(2,4),(2,5)(2,6),(3,1),(3,2),(3,3),(3,4),$

$(3,5)(3,6),(4,1),(4,2),(4,3),(4,4),(4,5)(4,6),(6,1),(6,2),$

$(6,3),(6,4),(6,5),(6,6)\}$
$\Rightarrow$ $n(A) = 2,$ $n(B) = 30$

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

and $P(B) = \frac{{30}}{{36}} = \frac{5}{6}$
$\Rightarrow$ $P(A) \cdot P(B) = \frac{5}{{108}}$

and $P(A \cap B) = \frac{1}{{36}} \ne P(A) \cdot P(B)$

So, A and B are not independent.