Question

# Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that (i) all the five cards are spades? (ii) only 3 cards are spades? (iii) none is a spade?

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Solution

## Let, X be the number of spades among the five drawn cards. Total number of cards in a deck is 52 and the total number of spaded in that deck is 52. In Bernoulli trials, p is the probability of success and q is the probability of failure. The probability of getting a spade is, p= 13 52 = 1 4 The probability of not getting spade is, q=1−p =1− 1 4 = 3 4 According to binomial distribution, the probability is calculated as, P( X=x )= C n n−x q n−x p x Here, n is total number of trials and x is number of successes. (i) The probability of getting all five cards of spade is, P( X=5 )= C 5 5 ( 3 4 ) 5−5 ⋅ ( 1 4 ) 5 = 1 1024 Thus, the probability of getting all 5 cards of spade is 1 1024 . (ii) The probability of getting only three cards of spade is, P( X=3 )= C 5 3 ( 3 4 ) 5−3 ⋅ ( 1 4 ) 3 = 5! 3!( 5−3 )! ( 3 4 ) 2 ⋅ ( 1 4 ) 3 =10⋅ 9 16 ⋅ 1 64 = 45 512 Thus, the probability of getting only 3 cards of spade is 45 512 . (iii) The probability of getting no spade is, P( X=0 )= C 5 0 ( 3 4 ) 5−0 ⋅ ( 1 4 ) 0 = 5! 0!( 5−0 )! ( 3 4 ) 5 ⋅( 1 ) = 243 1024 Thus, the probability of getting no spade is 243 1024 .

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