Question

Two cards are drawn (without replacement) from a well shuffled deck of $52$ cards. Find probability distribution and mean of number of cards numbered $4$

Answer 1

Total no. of cards in the pack $=52$

The number of cards numbered $4$ in the pack of cards $=4$

Let $X$ be the discrete random variable denoting the number of cards numbered $4$ when two cards are drawn without replacement.

$X$ takes the values $0,1,2$.

Now,

Case I:- Probability of no $4$ being drawn

$P(X=0)=\frac{{}^{48}{C}_2}{{}^{52}{C}_2}=\frac{188}{221}$

Case II:- Probability of one $4$ being drawn

$P(X=1)=\frac{{}^{48}{C}_1\times {}^4{C}_1}{{}^{52}{C}_2}=\frac{32}{221}$

Case III:- Probability of two $4$ being drawn

$P(X=2)=\frac{{}^4{C}_2}{{}^{52}{C}_2}=\frac{1}{221}$

Therefore, the probability distribution is-

$X$	$0$	$1$	$2$
$P\left(x\right)$	$\frac{188}{221}$	$\frac{32}{221}$	$\frac{1}{221}$

Now, mean of probability distribution-

Mean $=\sum x_i{P}_i=0\times \frac{188}{221}+1\times \frac{32}{221}+2\times \frac{1}{221}=\frac{0+32+2}{221}=\frac{34}{221}$