Question

Four cards are drawn successively with replacement from a well shuffled deck of $52$ cards. What is the probability that.
(i) all the four cards are spades?
(ii) only $2$ cards are spades?

Answer 1

$4$ cards drawn successively with replacement from well shuffled deck of $52$ cards

$\left(1\right)$ if all four cards are spades

$1$ space occuring $\frac{13}{52}=\frac{1}{4}$

All four occuring $=\frac{1}{4}\times \frac{1}{4}\times \frac{1}{4}\times \frac{1}{4}=\frac{1}{256}$

$\left(2\right)$ If two cards are spades $=\frac{1}{4}\times \frac{1}{4}=\frac{1}{16}=\frac{1}{16}-\frac{1}{64}-\frac{1}{256}$

$\frac{1}{64}$ is probability of getting $3$ spades

$=\frac{16-4-1}{256}=\frac{11}{256}$