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Question
Mark the correct alternative in the following question:
The probability that a person is not a swimmer is 0.3. The probability that out of 5 persons 4 are swimmers is
Options
\[^{5}{}{C}_4 \left( 0 . 7 \right)^4 \left( 0 . 3 \right)\]
\[^{5}{}{C}_1 \left( 0 . 7 \right) \left( 0 . 3 \right)^4\]
\[^{5}{}{C}_4 \left( 0 . 7 \right) \left( 0 . 3 \right)^4\]
\[\left( 0 . 7 \right)^4 \left( 0 . 3 \right)\]
Solution
\[\text{ We have,} \]
\[q = \text{ probability that a person is not a swimmer = 0 . 3 and } \]
\[\text{ p = probability that a person is a swimmer} = 1 - q = 1 - 0 . 3 = 0 . 7\]
\[\text{ Let X denote a success that a person selected is a swimmer . Then, } \]
\[\text{ X follows the binomial distribution with parameters n = 5 and } p = 0 . 7\]
\[ \therefore P\left( X = r \right) = ^{5}{}{C}_r p^r q^\left( 5 - r \right) = ^{5}{}{C}_r \left( 0 . 7 \right)^r \left( 0 . 3 \right)^\left( 5 - r \right) \]
\[\text{ Now,} \]
\[\text{ The required probability } = P\left( X = 4 \right) = ^{5}{}{C}_4 \left( 0 . 7 \right)^4 \left( 0 . 3 \right)^\left( 5 - 4 \right) =^{5}{}{C}_4 \left( 0 . 7 \right)^4 \left( 0 . 3 \right)\]
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