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’.

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.