Question

Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.

Answer 1

Let X denote the number of hearts in a sample of 3 cards drawn from a well-shuffled deck of 52 cards. Then, X can take the values 0, 1, 2 and 3.
Now,

\[P\left( X = 0 \right)\]
\[ = P\left( \text{ no heart } \right)\]
\[ = \frac{39}{52} \times \frac{39}{52} \times \frac{39}{52}\]
\[ = \frac{27}{64}\]
\[P\left( X = 1 \right)\]
\[ = P\left( 1 \text{ heart } \right)\]
\[ = \left( \frac{13}{52} \times \frac{39}{52} \times \frac{39}{52} \right) + \left( \frac{39}{52} \times \frac{13}{52} \times \frac{39}{52} \right) + \left( \frac{39}{52} \times \frac{39}{52} \times \frac{13}{52} \right)\]
\[ = \frac{27}{64}\]
\[P\left( X = 2 \right)\]
\[ = P\left( 2 \text{ hearts } \right)\]
\[ = \left( \frac{13}{52} \times \frac{13}{52} \times \frac{39}{52} \right) + \left( \frac{39}{52} \times \frac{13}{52} \times \frac{13}{52} \right) + \left( \frac{13}{52} \times \frac{39}{52} \times \frac{13}{52} \right)\]
\[ = \frac{9}{64}\]
\[P\left( X = 3 \right)\]
\[ = P\left( 3 \text{ hearts } \right)\]
\[ = \frac{13}{52} \times \frac{13}{52} \times \frac{13}{52}\]
\[ = \frac{1}{64}\]

Thus, the probability distribution of X is given by

X	P(X)
0	\[\frac{27}{64}\]
1	\[\frac{27}{64}\]
2	\[\frac{9}{64}\]
3	\[\frac{1}{64}\]