Question

Two cards are drawn 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,

\[P\left( X = 0 \right)\]
\[ = P\left( \text{ no ace }\right)\]
\[ = \frac{{}^{48} C_2}{{}^{52} C_2}\]
\[ = \frac{2256}{2652}\]
\[ = \frac{188}{221}\]
\[P\left( X = 1 \right)\]
\[ = P\left( 1 \text{ ace } \right)\]
\[ = \frac{{}^4 C_1 \times^{48} C_1}{{}^{52} C_2}\]
\[ = \frac{192}{1326}\]
\[ = \frac{32}{221}\]
\[P\left( X = 2 \right)\]
\[ = P\left( 2 \text{ aces}  \right)\]
\[ = \frac{{}^4 C_2}{{}^{52} C_2}\]
\[ = \frac{6}{1326}\]
\[ = \frac{1}{221}\]

Thus, the probability distribution of X is given by

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