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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=1p =1 1 4 = 3 4

According to binomial distribution, the probability is calculated as,

P( X=x )= C n nx q nx 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 ) 55 ( 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 ) 53 ( 1 4 ) 3 = 5! 3!( 53 )! ( 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 ) 50 ( 1 4 ) 0 = 5! 0!( 50 )! ( 3 4 ) 5 ( 1 ) = 243 1024

Thus, the probability of getting no spade is 243 1024 .


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