Question

Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the mean and variance of number of red cards.

Answer 1

It is given that the cards are drawn successively with replacement so the the events are independent. Therefore, the drawing of the cards follow binomial distribution.

​Probability of drawing a red card = p = $\frac{26}{52}=\frac{1}{2}$

∴ q = 1 − p = $1-\frac{1}{2}=\frac{1}{2}$

Also, n = 3

Let X be the random variable denoting the number of red cards drawn from a well shuffled pack of 52 cards.

$\therefore P\left(X=r\right)={}^{3}C_r{\left(\frac{1}{2}\right)}^{3-r}{\left(\frac{1}{2}\right)}^r={}^{3}C_r{\left(\frac{1}{2}\right)}^3,r=0,1,2,3$

Probability of drawing no red ball = $P\left(X=0\right)={}^3\mathrm{C}_0{\left(\frac{1}{2}\right)}^3=\frac{1}{8}$
Probability of drawing one red ball = $P\left(X=1\right)={}^3\mathrm{C}_1{\left(\frac{1}{2}\right)}^3=\frac{3}{8}$
Probability of drawing two red balls = $P\left(X=2\right)={}^3\mathrm{C}_2{\left(\frac{1}{2}\right)}^3=\frac{3}{8}$
Probability of drawing three red balls = $P\left(X=3\right)={}^3\mathrm{C}_3{\left(\frac{1}{2}\right)}^3=\frac{1}{8}$

Thus, the probability distribution of X is as follows:

$x_i$	$p_i$	$p_i{x}_i$	$p_i{x}_i^2$
0	$\frac{1}{8}$	0	0
1	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{3}{8}$
2	$\frac{3}{8}$	$\frac{6}{8}$	$\frac{12}{8}$
3	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{9}{8}$
		$\sum _{}^{}{p}_i{x}_i=\frac{12}{8}$	$\sum _{}^{}{p}_i{x}_i^2=3$

Mean of X $=\sum _{}^{}{p}_i{x}_i=\frac{12}{8}=\frac{3}{2}$

Variance of X $=\sum _{}^{}{p}_i{x}_i^2-{\left(\mathrm{Mean}\right)}^2=3-\frac{9}{4}=\frac{3}{4}$