Question

Three cards are drawn successively with replacement from a well shuffled pack of $52$ cards. Find the probability distribution of the number of spades. Hence find the mean of the distribution.

Answer 1

$0$ spades	$1$ spades	$2$ spades	$3$ spades	Total
$\frac{3}{4}\times \frac{3}{4}\times \frac{3}{4}=\frac{27}{64}$	$3\times \frac{1}{4}\times \frac{3}{4}\times \frac{3}{4}=\frac{27}{64}$	$3\times \frac{1}{4}\times \frac{1}{4}\times \frac{3}{4}=\frac{9}{64}$	$\frac{1}{4}\times \frac{1}{4}\times \frac{1}{4}=\frac{1}{64}$	$\frac{27}{64}+\frac{27}{64}+\frac{9}{64}+\frac{1}{64}=\frac{64}{64}=1$

The probability of getting a spade remains the same in every pick because the cards are replaced after each pick thus making the picks independent events.

The mean of the distribution is $\sum x_{i}P(X=x_i)=\frac{3}{4}$.