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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?

Q. 4   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?

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Let X represent a number of spade cards among five drawn cards. Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards.

We have 13 spades.

        P=\frac{13}{52}=\frac{1}{4}

        q=1-P=1-\frac{1}{4}=\frac{3}{4}

X has a binomial distribution,n=5.

\therefore \, \, \, \, P(X=x)=^nC_x.q^{n-x}.p^x

                P(X=x)=^5C_x.(\frac{3}{4})^{5-x} . (\frac{1}{4})^{x}

             Put X=5 ,

                   P(X=5)=^5C_5.(\frac{3}{4})^{0} . (\frac{1}{4})^{5}

                                          =1\times \frac{1}{1024}

                                         = \frac{1}{1024}                     

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