#### Question

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes and, hence, find its mean.

#### Solution

Let *X* be the number of times a doublet is obtained in four throws.

Then, *p* = probability of success in one throw of a pair of dice =

\[\frac{6}{36} = \frac{1}{6}\]

\[\text{ and } q = \frac{5}{6}; n = 4\]

\[P(X = r) = ^ {4}{}{C}_r \left( \frac{1}{6} \right)^r \left( \frac{5}{6} \right)^{4 - r} , r = 0, 1, 2, 3, 4\]

\[\text{ As n = 4 and } p = \frac{1}{6}, \]

\[\text{ mean } = np = \frac{4}{6} = \frac{2}{3}\]

\[\therefore P(X = r) =^ {4}{}{C}_r \left( \frac{5}{6} \right)^r \left( \frac{1}{6} \right)^{n - r} , r = 0, 1, 2, 3, 4\]

\[\text{ The distribution is as follows: } \]

X 0 1 2 3 4

\[P(X) \left( \frac{5}{6} \right)^4 \frac{20}{6^4} \frac{150}{6^4} \frac{500}{6^4} \frac{1}{6^4}\]