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?
(i)E:'the card drawn 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'

Answer 1

$S$={All the 52 cards}⇒$n(S)=52$
(i)$E$={13 spades}⇒$n(E)=13$
$∴P(E)=\frac{n(E)}{n(S)}=\frac{13}{52}=\frac{1}{4}$
$F$={4 aces}$⇒n(F)=4$
$∴P(F)=\frac{n(F)}{n(S)}=\frac{4}{52}=\frac{1}{13}$
Now $E∩F=${An ace of spade}$⇒n(E∩F)=1$
$∴P(E∩F)=\frac{n(E∩F)}{n(S)}=\frac{1}{52}$
Also.$P(E).P(F)=\frac{1}{4}×\frac{1}{13}=\frac{1}{52}$
Therefore,$P(E∩F)=P(E).P(F)$
Hence,$E$ and $F$ are independent events.
(ii)$E$={26 black cards}$⇒n(E)=26$
$∴P(E)=\frac{n(E)}{n(S)}=\frac{26}{52}=\frac{1}{2}$
$F=${4 kings}$⇒n(F)=4$
$∴P(F)=\frac{n(F)}{n(S)}=\frac{4}{52}=\frac{1}{13}$
Now,$E∩F=${2 black kings}$⇒n(E∩F)=2$
$∴P(E∩F)=\frac{n(E∩F)}{n(S)}=\frac{2}{52}=\frac{1}{26}$
Also,$P(E).P(F)=\frac{1}{2}×\frac{1}{13}=\frac{1}{26}$
Therefore,$P(E∩F)=P(E).P(F)$
Hence,$E$ and $F$ are independent events.
(iii)$E$={4kings,4queens}$⇒n(E)=8$
$∴P(E)=\frac{n(E)}{n(S)}=\frac{8}{52}=\frac{2}{13}$
$F$={4queens,4jacks}$⇒n(F)=8$
$∴P(F)=\frac{n(F)}{n(S)}=\frac{8}{52}=\frac{2}{13}$
Now $E∩F=${4queens}$⇒n(E∩f)=4$
$∴P(E∩F)=\frac{n(E∩F)}{n(S)}=\frac{4}{52}=\frac{1}{13}$
Also,$P(E).P(F)=\frac{2}{13}×\frac{2}{13}=\frac{4}{169}$
Therefore,$P(E∩F)≠P(E).P(F)$
Hence,$E$ and $F$ are not independent events.