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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,

The probability of not getting spade is,

According to binomial distribution, the probability is calculated as,

Here, *n* is total number of trials and *x* is number of successes.

(i)

The probability of getting all five cards of spade is,

Thus, the probability of getting all 5 cards of spade is

(ii)

The probability of getting only three cards of spade is,

Thus, the probability of getting only 3 cards of spade is

(iii)

The probability of getting no spade is,

Thus, the probability of getting no spade is

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