Question

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards, find the probability that 

(i) all the five cards are spades 

(ii) only 3 cards are spades 

(iii) none is a spade.

Answer 1

Let X = number of spade cards. 

p = probability of drawing a spade card from a pack of 52 cards. 

Since there are 13 spade cards in the pack of 52 cards.

The p.m.f. of X is given by

(i) P(all five cards are spade)

Hence, the probability of all the five cards are spades = \(\frac{1}{1024}\)

(ii) P(only 3 cards are spade) = P[X = 3]

Hence, the probability of only 3 cards are spades = \(\frac{45}{512}\)

(iii) P(none of cards is spade) = P[X = 0]

Hence, the probability of none of the cards is a spade = \(\frac{243}{1024}\)