Question

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

Answer 1

The number of aces in the deck of card$=4$

suppose $x$ is the discrete

$p\left(x=0\right)=$ probability of no aces being drawn

$\begin{array}{c}=\frac{48c_2}{52c_2}\\ =\frac{\frac{48\times 47}{1\times 2}}{\frac{52\times 51}{1\times 2}}\\ =\frac{564}{663}\end{array}$

$p\left(x=1\right)=$ probability that one card is a aces

$\begin{array}{c}=\frac{48c_1\times 4c_1}{52c_2}\\ =\frac{48\times 4}{\frac{52\times 51}{1\times 2}}\\ =\frac{96}{663}\\ =\frac{32}{221}\end{array}$

$p\left(x=2\right)=$ probability that both cards are aces

$\begin{array}{c}=\frac{4c_2}{52c_2}\\ =\frac{\frac{4\times 3}{1\times 2}}{\frac{52\times 51}{1\times 2}}\\ =\frac{3}{663}\\ =\frac{1}{221}\end{array}$