#### 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.

∴ P(E) = P(the card drawn is a spade) =`12/52 = 1/4`

∴ P(F) = P(the card drawn is an ace) =`4/52 = 1/13`

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

P(EF) = P(the card drawn is spade and an ace) =`1/52`

P(E) × P(F) =`1/4. 1/13 = 1/52 = P(EF)`

⇒ P(E) × P(F) = P(EF)

Therefore, the events E and F are independent.

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

∴ P(E) = P(the card drawn is black) =`26/52 = 1/2`

∴ P(F) = P(the card drawn is a king) =`4/52 = 1/13`

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

∴ P (EF) = P(the card drawn is a black king) =`2/52 = 1/26`

P(E) × P(F) =`1/2. 1/3 = 1/26 = P(EF)`

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.

∴ P(E) = P(the card drawn is a king or a queen) =`8/52 = 2/13`

∴ P(F) = P(the card drawn is a queen or a jack) =`8/52 = 2/13`

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

∴ P(EF) = P(the card drawn is a king or a queen, or queen or a jack)

=`4/52 = 1/13`

P(E) × P(F) =`2/13. 2/13= 4/169 != 1/13`

`=> P(E).P(F) != P(EF)`

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