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 of card is with replacement, the trials are Bernoulli trials.

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

`Rightarrow p=frac{13}{52}=frac{1}{4} `

`therefore q=1-frac{1}{4}=frac{3}{4}`

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

`{P}({X}=x) =^{n} {C}_{x} q^{n-x} p^{x}, text { where } x=0,1, ldots n `

`=^{5} {C}_{x}(frac{3}{4})^{5-x}(frac{1}{4})^{x}`

`(i) P `(all five cards are spades )`=P(X=5)`

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