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?

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’

Solution:

(i) In a deck of 52 cards, 13 cards are spades and 4 cards are aces.

$\therefore P(E)=P($ the card drawn is a spade $)=\frac{13}{52}=\frac{1}{4}$

$\therefore \mathrm{P}(\mathrm{F})=\mathrm{P}$ (the card drawn is an ace) $=\frac{4}{52}=\frac{1}{13}$

In the deck of cards, only 1 card is an ace of spades.

$P(E F)=P($ the card drawn is spade and an ace $)=\frac{1}{52}$

$\Rightarrow P(E) \times P(F)=P(E F)$

Therefore, the events E and F are independent.

(ii) In a deck of 52 cards, 26 cards are black and 4 cards are kings.

$\therefore P(E)=P($ the card drawn is black $)=\frac{26}{52}=\frac{1}{2}$

$\therefore P(F)=P($ the card drawn is a king $)=\frac{4}{52}=\frac{1}{13}$

In the pack of 52 cards, 2 cards are black as well as kings.

$\therefore \mathrm{P}(\mathrm{EF})=\mathrm{P}($ the card drawn is a black king $)=\frac{2}{52}=\frac{1}{26}$

$P(E) \times P(F)=\frac{1}{2} \cdot \frac{1}{13}=\frac{1}{26}=P(E F)$

Therefore, the given 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.

$\therefore P(E)=P($ the card drawn is a king or a queen $)=\frac{8}{52}=\frac{2}{13}$

$\therefore \mathrm{P}(\mathrm{F})=\mathrm{P}($ the card drawn is a queen or a jack $)=\frac{8}{52}=\frac{2}{13}$

There are 4 cards which are king or queen and queen or jack.

$\therefore P(E F)=P($ the card drawn is a king or a queen, or queen or a jack)

$=\frac{4}{52}=\frac{1}{13}$

$P(E) \times P(F)=\frac{2}{13} \cdot \frac{2}{13}=\frac{4}{169} \neq \frac{1}{13}$

$\Rightarrow \mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F}) \neq \mathrm{P}(\mathrm{EF})$

Therefore, the given events E and F are not independent.

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