Question

A card is thrown from a pack of $52$ cards so that each card equally likely to be selected. Then find whether the events $A$ and $B$ independent?

$A=$the card drawn is spade, $B=$the card drawn in an ace.

Answer 1

In a pack of $52$ cards there are $13$ spade,$4$ ace

$A=$the card drawn is a spade.

$P\left(A\right)=\frac{13}{52}=\frac{1}{4}$

$B=$the card drawn is an ace

$P\left(B\right)=\frac{4}{52}=\frac{1}{13}$

$A\cap B=$Card is an ace from spade

$P(A\cap B)=\frac{1}{52}$

$P\left(A\right)P\left(B\right)=\frac{1}{4}\times \frac{1}{13}=\frac{1}{52}$

$P\left(A\right)P\left(B\right)=P(A\cap B)$

$\therefore A$ and $B$ are independent events.