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# 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? - CBSE (Science) Class 12 - Mathematics

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#### 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.

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Solution 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? Concept: Independent Events.
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