Question

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

Answer 1

Let X denote the number of aces in a sample of 2 cards drawn from a well-shuffled pack of 52 playing cards. Then, X can take the values 0, 1 and 2.
Now,
$$\begin{gathered} P\left(X=0\right) \\ =P\left(\mathrm{no}\mathrm{ace}\right) \\ =\frac{48}{52}\times \frac{47}{51} \\ =\frac{2256}{2652} \\ =\frac{188}{221} \\ P\left(X=1\right) \\ =P\left(1\mathrm{ace}\right) \\ =\frac{4}{52}\times \frac{48}{51}+\frac{48}{52}\times \frac{4}{51} \\ =\frac{384}{2652} \\ =\frac{32}{221} \\ P\left(X=2\right) \\ =P\left(2\mathrm{aces}\right) \\ =\frac{4}{52}\times \frac{3}{51} \\ =\frac{12}{2652} \\ =\frac{1}{221} \end{gathered}$$

Thus, the probability distribution of X is given by

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