Question

Five cards are drawn successively with replacement from a well-shuffled pack 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?

Answer 1

Let X denote the number of spade cards when 5 cards are drawn with replacement. Because it is with replacement,

X follows a binomial distribution with n = 5; $p\mathit{}=\frac{13}{52}=\frac{1}{4};\mathit{}q=1-p=\frac{3}{4}$

$\mathrm{P}(X=r)={}^{5}C_r{\left(\frac{1}{4}\right)}^r{\left(\frac{3}{4}\right)}^{5-r}$

$$\begin{gathered} \left(\mathrm{i}\right)P(\mathrm{All}\mathrm{cards}\text{are}\mathrm{spades})\hspace{0.17em}=P(X=5) \\ ={}^{5}C_5{\left(\frac{1}{4}\right)}^5{\left(\frac{3}{4}\right)}^0 \\ =\frac{1}{1024} \\ \left(\mathrm{ii}\right)P(\mathrm{only}3\mathrm{cards}\text{are}\mathrm{spades})=P(X=3) \\ ={}^{5}C_3{\left(\frac{1}{4}\right)}^3{\left(\frac{3}{4}\right)}^2 \\ =\frac{1}{1024}\left(90\right) \\ =\frac{45}{512} \\ \left(\mathrm{iii}\right)P(\mathrm{none}\mathrm{is}\text{a}\mathrm{spade})\hspace{0.17em}=\mathit{}P(X=0) \\ ={}^{5}C_0{\left(\frac{1}{4}\right)}^0{\left(\frac{3}{4}\right)}^5 \\ =\frac{243}{1024} \end{gathered}$$