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,
$$\begin{gathered} P\left(X=0\right) \\ =P\left(\mathrm{no}\mathrm{heart}\right) \\ =\frac{39}{52}\times \frac{39}{52}\times \frac{39}{52} \\ =\frac{27}{64} \\ P\left(X=1\right) \\ =P\left(1\mathrm{heart}\right) \\ =\left(\frac{13}{52}\times \frac{39}{52}\times \frac{39}{52}\right)+\left(\frac{39}{52}\times \frac{13}{52}\times \frac{39}{52}\right)+\left(\frac{39}{52}\times \frac{39}{52}\times \frac{13}{52}\right) \\ =\frac{27}{64} \\ P\left(X=2\right) \\ =P\left(2\mathrm{hearts}\right) \\ =\left(\frac{13}{52}\times \frac{13}{52}\times \frac{39}{52}\right)+\left(\frac{39}{52}\times \frac{13}{52}\times \frac{13}{52}\right)+\left(\frac{13}{52}\times \frac{39}{52}\times \frac{13}{52}\right) \\ =\frac{9}{64} \\ P\left(X=3\right) \\ =P\left(3\mathrm{hearts}\right) \\ =\frac{13}{52}\times \frac{13}{52}\times \frac{13}{52} \\ =\frac{1}{64} \end{gathered}$$

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	$\frac{1}{64}$