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?

Let the number of spade cards among the five drawn cards be x.

As we can observe that the drawing of cards is with replacement, thus, the trials will be Bernoulli trials.

Now, we know that in a deck of 52 cards there are total 13 spade cards.

Thus, Probability of drawing a spade from a deck of 52 cards

Thus, x has a binomial distribution with

Thus, P(X = x) = ^{n}C_{x}q^{n-x}p^{x} , where x = 0, 1, 2, …n

(i) Probability of drawing all five cards as spades = P(X = 5)

(ii) Probability of drawing three out five cards as spades = P(X = 3)

(iii) Probability of drawing all five cards as non-spades = P(X = 0)

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