Question

Two cards are drawn successively without replacement from a well-shuffled deck of $52$ card . Find the probability distribution of the number of aces.

Answer 1

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

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

$X$ takes the value $0,1$ and $2$.

Case I:- Probability of no queen being drawn, i.e., $X=0$,

$P(X=0)=\frac{{}^{48}{C}_2\times {}^4{C}_0}{{}^{52}{C}_2}$

As we know that,

${}^n{C}_r=\frac{n!}{r!(n-r)!}$

$\therefore P(X=0)=\frac{1128\times 1}{1326}=\frac{188}{221}$

Case I:- Probability of $1$ queen being drawn, i.e., $X=1$,

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

$\Rightarrow P(X=1)=\frac{48\times 4}{1326}=\frac{32}{221}$

Case III:- Probability of $2$ queen being drawn, i.e., $X=2$,

$P(X=2)=\frac{{}^{48}{C}_0\times {}^4{C}_2}{{}^{52}{C}_2}$

$\Rightarrow P(X=2)=\frac{1\times 6}{1326}=\frac{1}{221}$

Therefore, the probability distribution will nbe-

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