Question

One card is drawn at random from a well- shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
E : the card drawn is black, F: the card drawn is a king

Answer 1

In a deck of 52 cards, 26 cards are black and 4 cards are kings. Given, E : the cards drawn is black $\Rightarrow n\left(E\right)=26$
F: the cards drawn is kng $\Rightarrow n\left(F\right)=4$
Also, n(S) = 52
P(E) = P (cards drawn is black) = $\frac{n\left(E\right)}{n\left(S\right)}=\frac{26}{52}=\frac{1}{2}$
and P(F) = P (card drawn is a king) $\frac{n\left(E\right)}{n\left(S\right)}=\frac{4}{52}=\frac{1}{13}$
Also, $E\cup F:$ card drawn is a black king
$\Rightarrow P(E\cap F)=\frac{n(E\cap S)}{n\left(S\right)}=\frac{2}{52}=\frac{1}{26}Now,p\left(E\right)\times P\left(F\right)=\frac{1}{2}\times \frac{1}{13}=\frac{1}{26}P(E\cap F)\Rightarrow P(E\cap F)=P\left(E\right)P\left(F\right)$