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,

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

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	$$\frac{1}{64}$$