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Question

Five cards are drawn successively with replacement from well- shuffled deck of 52 cards. What is the probability that

(ii) only 3 cards are spades?

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Solution

Let X represent the number of spade cards among the five cards drawn. Since, the drawing card is with replacement, the trials are Bernoulli trails.

In a well-shuffled deck of 52 cards, there are 13 spade cards.

p=P(success) = P (a spade card is drawn) =1352=14 and q=1- 14=34.

X has a binomial distribution with n=5, p= 14 and q=34

Therefore, P(X=r)= nCrprqnr,where r=0,1,2,..., n

P(X=r) =5Cr(14r)(34)5r (By Binomial distribution)

P (only three cards are spades) = P(X=5)

=5C3p3q2 = 5.4.33!(14)3(34)2=601×2×3×3245=901024=45512


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