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