One card is drawn at random from a well shumed dech of 52 cards. In which of the following cases are the events $E$ and $F$ independent?

(i) $E$ : `the car4 Brawn is a spade'
$F$ : `the card drawn is an ace'

(ii) $E$ : `the card drawn is black'
$F$ : `the card drawn is a king'

(iii) $E$ : `the card drawn is a king or queen'

$F$ : `the card drawn is a queen or jack'.

.: (i) $P(E) = \cfrac{{13}}{{52}} = \cfrac{1}{4}$

and $P(F) = \cfrac{4}{{52}} = \cfrac{1}{{13}}$

$\Rightarrow$ $P(E \cap F) = \cfrac{1}{{52}} = \cfrac{1}{4} \times \cfrac{1}{{13}} = P(E)P(F)$

m $\Rightarrow$ $E$ and $F$ are independent.

(ii) $P(E) = \cfrac{{26}}{{52}} = \cfrac{1}{2}$

and
$P(F) = \cfrac{4}{{52}} = \cfrac{1}{{13}}$ $\Rightarrow$ $P(E \cap F) = \cfrac{2}{{52}} = \cfrac{1}{{26}} = \cfrac{1}{2} \times \cfrac{1}{{13}}$

$\Rightarrow$ $P(E \cap F) = P(E)P(F)$ $\Rightarrow$ $E$

and $F$ are independent.

(iii) $P(E) = \cfrac{4}{{52}} + \cfrac{4}{{52}} = \cfrac{{4 + 4}}{{52}} = \cfrac{8}{{52}} = \cfrac{2}{{13}}$

and
$P(F) = \cfrac{4}{{52}} + \cfrac{4}{{52}} = \cfrac{{4 + 4}}{{52}} = \cfrac{8}{{52}} = \cfrac{2}{{13}}$

$\Rightarrow$ $P(E \cap F) = \cfrac{4}{{52}} = \cfrac{1}{{13}} \ne \cfrac{2}{{13}} \times \cfrac{2}{{13}}$ $\Rightarrow$ $P(E \cap F) \ne P(E)P(F)$

$\Rightarrow$ $E$ and $F$ are not independent.