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?

Explanation

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


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

p = P (success) = P (a spade card is drawn) = `13/52 = 1/4`

and q = `1 - p = 1 - 1/4 = 3/4`

X has a binomial distribution with `n = 5, p = 1/4 and q = 3/4`

Therefore, by Binomial distribution `P(X = r) = n^C^r p^rq^n – r, where r = 0, 1, 2,...,n`

`P(X = r) = ^C5^r(1/4)^r(3/4)^5 - r`

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