Question

Five cards are drawn successively with replacement from well- shuffled deck of 52 cards. What is the probability that

(ii) only 3 cards are spades?

Answer 1

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

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

p=P(success) = P (a spade card is drawn) =$\frac{13}{52}=\frac{1}{4}$ and q=1- $\frac{1}{4}=\frac{3}{4}$.

X has a binomial distribution with n=5, p= $\frac{1}{4}andq=\frac{3}{4}$

Therefore, P(X=r)= ${}^n{C}_r{p}^r{q}^{n-r},$where r=0,1,2,..., n

P(X=r) =${}^5{C}_r\left({\frac{1}{4}}^r\right){\left(\frac{3}{4}\right)}^{5-r}$ (By Binomial distribution)

P (only three cards are spades) = P(X=5)

=${}^5{C}_3{p}^3{q}^2$ = $\frac{\mathrm{5.4.3}}{3!}{\left(\frac{1}{4}\right)}^3{\left(\frac{3}{4}\right)}^2=\frac{60}{1\times 2\times 3}\times \frac{3^2}{4^5}=\frac{90}{1024}=\frac{45}{512}$