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 spade?
(ii) only 3 cards are spades?
(iii) none of spade?

Answer 1

Let P=probability of a success and q=probability of a failure
$S=52 \, cards \Rightarrow (S) = 52\, and \, A= {13 spades} \Rightarrow n(A) =13$
$p= \frac{n(A)}{n(S)}=\frac{13}{52}=\frac{1}{4}\, and \, q =1-\frac{1}{4}=\frac{3}{4}$
(i)n=5,r=5
P(X=r)=$^nC_rp^rq^{n-r}$
P(X=5)=$^5C_{5}\bigg(\frac{1}{4}\bigg)^5=\frac{1}{1024}$

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(ii)n=5,r=0
P(X=3)$^5C_3\bigg(\frac{1}{4}\bigg)^3\bigg(\frac{3}{4}\bigg)^2=\frac{5*4*3}{1*2*3*9}\bigg(\frac{1}{4}\bigg)^5=\frac{90}{1024}=\frac{45}{512}$

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(iii)n=5,r=0
P(X=0)=$^5C_0.\bigg(\frac{3}{4}\bigg)^5=\frac{243}{1024}$