# Three cards are drawn at random (without replacement) from a well shuffled pack of 52 playing cards. Find the probability distribution of number of red cards. Hence find the mean of the distribution.

Toolbox:
• Mean of the probability distribution $=\overline {X}= \sum (X_i \times P(X_i))$
Three Cards are drawn from a pack of 52 cards.
Let X be the number of red cards drawn.
In the pack of 52 cards there are 26 red cards and 26 black cards.
We can draw 3 cards out of 52 in $^{52}C_3$ ways.
P (X = 0) = P (no red cards drawn) = $\large\frac{^26C_0.^{26}C_3}{ ^{52}C_3} =\frac{2}{17}$
P (X = 1) = P (one red card drawn) = $\large\frac{^{26}C_1.^{26}C_2}{ ^{52}C_3} =\frac{13}{34}$
P (X = 1) = P (two red card drawn) = $\large\frac{^{26}C_2.^{26}C_1}{ ^{52}C_3} =\frac{13}{34}$
P (X = 0) = P (no red cards drawn) = $\large\frac{^26C_3.^{26}C_0}{ ^{52}C_3} =\frac{2}{17}$
$\therefore\:\:$ Probability distribution of number of red cards is given by
 $x_i$ 0 1 2 3 $P(x_i)$ $\large\frac{2}{17}$ $\large\frac{13}{34}$ $\large\frac{13}{34}$ $\large\frac{2}{17}$ $x.P(x_i)$ 0 $\large\frac{13}{34}$ $\large\frac{26}{34}$ $\large\frac{6}{17}$ $\sum x_i.P(x_i)=\large\frac{3}{2}$

$\Rightarrow\:$  Mean of the distribution is $=\sum x_i.P(x_i)=\large\frac{3}{2}$

edited Mar 23, 2014