**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 $=\sum x_i.P(x_i)=\large\frac{3}{2}$