Question

One card is drawn at random from a well-shufflec deck of 52 cards.
(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'
Which of the following option is correct?

• A. $\ln$ (i) $E$ and $F$ are independent
• B. $\ln$ (ii) $E$ and $F$ are independent
• C. In (iii) $E$ and $F$ are not independent
• D. All of the above

Answer 1

Answer: (D)

In a deck of 52 cards, 13 cards are spades and 4 cards are aces.
Given, $E$ : the card drawn is a spade
$\Rightarrow n(E)=13$
and $F$ : the card drawn is an ace
$n(F)=4\text{ and }n(S)=52$
Here, $P(E)=P$ (card drawn is spade $)$
$=\frac{n(E)}{n(S)}=\frac{13}{52}=\frac{1}{4}$
$P(F)=P($ card drawn is an ace $)=\frac{n(F)}{n(S)}=\frac{4}{52}=\frac{1}{13}$
Also, $E\cap F$ : the deck of cards, only 1 card is an ace of spades.
$\Rightarrow P(E\cap F)=\frac{n(E\cap F)}{n(S)}=\frac{1}{52}$
Now, $P(E)\times P(F)=\frac{1}{4}\times \frac{1}{13}=\frac{1}{52}=P(E\cap F)$
Therefore, the events $E$ and $F$ are independent.
(ii) In a deck of 52 cards, 26 cards are black and 4 cardsare kings.
Given, $E$ : the card drawn is black
$\Rightarrow n(E)=26$
$F$ : the card drawn is king
$\Rightarrow n(F)=4$
$\text{ Also, }n(S)=52$
$P(E)=P$ (card drawn is black)
$=\frac{n(E)}{n(S)}=\frac{26}{52}=\frac{1}{2}$
and $P(F)=P($ card drawn is a king $)$
$=\frac{n(F)}{n(S)}=\frac{4}{52}=\frac{1}{13}$
Also, $E\cap F$ : card drawn is a black king
$\Rightarrow P(E\cap F)=\frac{n(E\cap F)}{n(S)}=\frac{2}{52}=\frac{1}{26}$
$\text{ Now, }P(E)\times P(F)=\frac{1}{2}\times \frac{1}{13}=\frac{1}{26}=P(E\cap F)$
$\Rightarrow P(E\cap F)=P(E)P(F)$
Therefore, the events $E$ and $F$ are independent.
(iii) In a deck of 52 cards, 4 cards are kings, 4 cards are queens and 4 cards are jacks.
$P(E)=P$ (card drawn is a king or a queen)
$=P($ King $)+P($ Queen $)=\frac{4}{52}+\frac{4}{52}=\frac{8}{52}=\frac{2}{13}$
and $P(F)=P$ (cards drawn is a queen or a jack)
$=P($ Queen $)+P($ Jack $)=\frac{4}{52}+\frac{4}{52}=\frac{8}{52}=\frac{2}{13}$
Also, $E\cap F$ : card drawn is a queen
$\Rightarrow P(E\cap F)=\frac{4}{52}=\frac{1}{13}$
Now, $P(E)\times P(F)=\frac{2}{13}\times \frac{2}{13}$
$=\frac{4}{169}\ne P(E\cap F)$
$\Rightarrow P(F\cap F)\ne P(F)P(F)$
Therefore, the events $E$ and $F$ are not independent.