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

\[\text{ We have,} \]

\[q = \text{ probability that a person is not a swimmer = 0 . 3 and } \]

\[\text{ p = probability that a person is a swimmer} = 1 - q = 1 - 0 . 3 = 0 . 7\]

\[\text{ Let X denote a success that a person selected is a swimmer . Then, } \]

\[\text{ X follows the binomial distribution with parameters n = 5 and } p = 0 . 7\]

\[ \therefore P\left( X = r \right) = ^{5}{}{C}_r p^r q^\left( 5 - r \right) = ^{5}{}{C}_r \left( 0 . 7 \right)^r \left( 0 . 3 \right)^\left( 5 - r \right) \]

\[\text{ Now,} \]

\[\text{ The required probability } = P\left( X = 4 \right) = ^{5}{}{C}_4 \left( 0 . 7 \right)^4 \left( 0 . 3 \right)^\left( 5 - 4 \right) =^{5}{}{C}_4 \left( 0 . 7 \right)^4 \left( 0 . 3 \right)\]