Q. 15 4.0( 23 Votes )

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 :

Given: A deck of 52 cards.

(i) In a deck of 52 cards, 13 cards are spade and 4 cards are ace and only one card is there which is spade and ace both.


Hence, P(E) = The card drawn is a spade =


P(F) = The card drawn is an ace =


P(E F) = The card drawn is a spade and ace both = ..(1)


And P(E) . P(F)


..(2)


From (1) and (2)


P (E F) = P(E) . P(F)


Hence, E and F are independent events.


(ii) In a deck of 52 cards, 26 cards are black and 4 cards are king and only two card are there which are black and king both.


Hence, P(E) = The card drawn is of black =


P(F) = The card drawn is a king =


P(E F) = The card drawn is a black and king both = ..(1)


And P(E) . P(F)


..(2)


From (1) and (2)


P (E F) = P(E) . P(F)


Hence, E and F are independent events.


(iii) In a deck of 52 cards, 4 cards are queen, 4 cards are king and 4 cards are jack.


Hence, P(E) = The card drawn is either king or queen =


P(F) = The card drawn is either queen or jack =


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


P(E F) = The card drawn is either king or queen and either queen or jack = ..(1)


And P(E) . P(F)


..(2)


From (1) and (2)


P (E F) ≠ P(E) . P(F)


Hence, E and F are not independent events.


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