Question

Three cards are drawn at random (without replacement) from a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence, find the mean of the distribution. [CBSE 2014]

Answer 1

Let X denotes the number of red cards drawn.

Then, X can take the values 0, 1, 2 or 3.

Now,

$$\begin{gathered} \mathrm{P}\left(X=0\right)=\mathrm{P}\left(BBB\right)=\frac{26}{52}\times \frac{25}{51}\times \frac{24}{50}=\frac{2}{17}, \\ \mathrm{P}\left(X=1\right)=\mathrm{P}\left(RBB\mathrm{or}BRB\mathrm{or}BBR\right)=3\times \frac{26}{52}\times \frac{26}{51}\times \frac{25}{50}=\frac{13}{34}, \\ \mathrm{P}\left(X=2\right)=\mathrm{P}\left(RRB\mathrm{or}RBR\mathrm{or}BRR\right)=3\times \frac{26}{52}\times \frac{25}{51}\times \frac{26}{50}=\frac{13}{34}, \\ \mathrm{P}\left(X=3\right)=\mathrm{P}\left(RRR\right)=\frac{26}{52}\times \frac{25}{51}\times \frac{24}{50}=\frac{2}{17} \end{gathered}$$

Thus, the probability distribution of X is given by

X	P(X)
0	$\frac{2}{17}$
1	$\frac{13}{34}$
2	$\frac{13}{34}$
3	$\frac{2}{17}$

$$\begin{gathered} \mathrm{Mean}=\sum p_i{x}_i=0\times \frac{2}{17}+1\times \frac{13}{34}+2\times \frac{13}{34}+3\times \frac{2}{17} \\ =0+\frac{13}{34}+\frac{26}{34}+\frac{6}{17} \\ =\frac{51}{34} \\ =\frac{3}{2} \\ =1.5 \end{gathered}$$

Disclaimer: The answer given in the book is incorrect. The same has been corrected here.