Three cards are drawn with replacement from a standard deck. (a) Find the chance

Question

Three cards are drawn with replacement from a standard deck.

(a) Find the chance that all of the cards are face cards.

(b) Find the chance that at least one of the cards is a face card.

(c) Find the chance that none of the cards are face cards.

(d) Find the chance that not all of the cards are face cards.

(e) Find the chance that at least one of the cards is the king of hearts. Give answers as exact fractions after cancellation or to the nearest one-tenth of one percent. The face cards are the aces, kings, queens, and jacks.

Explanation / Answer

Total number of cards in a standard deck = 52 out of which,

face cards = 16 , heart = diamond = club = spade = 13 .

Let the face card be X

(a) P(all 3 cards are face cards) = (16/52)3 = 64/2197

(b) P(X>=1) = 1-P(X=0)

=1-(36/52)3

=1468/2197

(c) P(X=0) = ((52-16)/52)3 = 729/2197

(d) P(X is not equal to 3) = P(X=0)+P(X=1)+P(X=2)

= 729/2197 + 3C1(16/52)(36/52)2 + 3C2(16/52)2(36/52) = 2133/2197

(e) P( atleast one is king of hearts) = 1- P(no king of hearts)

= 1- (12/13)3 = 469/2197

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