Question

One card is drawn at random from shuffled well deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent?
(i)
$E:$ the card drawn is a spade
$F:$ the card drawn is a 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 jack

Answer 1

A and B are independent if $P(A\cap B)=p\left(A\right).P\left(B\right)$

$\left(1\right)p\left(E\right)=\frac{26}{52}=\frac{1}{2}$

$p\left(F\right)=\frac{4}{52}=\frac{1}{13}$

$p(E\cap F)=p$ (drawing black king) $=\frac{2}{52}=\frac{1}{26}$

$p(E\cap F)=p\left(E\right).p\left(F\right)$

$\frac{1}{26}=\frac{1}{2}\times \frac{1}{13}\Rightarrow$ E & F are independent.

$\left(2\right)p\left(E\right)=\frac{13}{52}=\frac{1}{4}$

$p\left(F\right)=\frac{4}{52}=\frac{1}{13}$

$p(E\cap F)=p$ (Ace of spades) $\frac{1}{52}$

$p(E\cap F)=p\left(E\right).p\left(F\right)$

$\frac{1}{52}=\frac{1}{4}\times \frac{1}{13}\Rightarrow$ E & F are independent.

$\left(3\right)p\left(E\right)=\frac{4}{52}+\frac{4}{52}=\frac{2}{13}$

$p\left(F\right)=\frac{4}{52}+\frac{4}{52}=\frac{2}{13}$

$p(E\cap F)=\frac{4}{52}$

$p(E\cap F)=p\left(E\right).p\left(F\right)$

$\frac{4}{52}\ne \frac{2}{13}\times \frac{2}{13}\Rightarrow$ E & F are not independent.