If the probability that a person is not a swimmer is 0.3, then the probability that out of 5 persons 4 are swimmers is

Options

A. `""^(5) C_(4) (0.7)^(0.4)`
B. `""^(5) C_(1) (0.7) (0.3)^(4)`
C. `""^(5) C_(4) (0.7) (0.3)^(4)`
D. `""^(5) C_(4) (0.7)^(4) (0.3)`

Answer

To solve the problem, we need to find the probability that out of 5 persons, 4 are swimmers, given that the probability that a person is not a swimmer is 0.3.

### Step-by-Step Solution:

1. **Identify the probabilities**:
- Let \( Q \) be the probability that a person is not a swimmer. Given \( Q = 0.3 \).
- Let \( P \) be the probability that a person is a swimmer. Therefore, \( P = 1 - Q = 1 - 0.3 = 0.7 \).

2. **Define the parameters for the binomial distribution**:
- We are conducting \( n = 5 \) trials (the number of persons).
- We want to find the probability of having \( r = 4 \) swimmers (successes).

3. **Use the binomial probability formula**:
The probability of getting exactly \( r \) successes in \( n \) trials is given by the formula:
\[ P(X = r) = \binom{n}{r} P^r Q^{n-r} \]
where \( \binom{n}{r} \) is the binomial coefficient.

4. **Calculate the binomial coefficient**:
\[ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5 \times 4!}{4! \times 1!} = 5 \]

5. **Substitute the values into the formula**:
Now we substitute \( n = 5 \), \( r = 4 \), \( P = 0.7 \), and \( Q = 0.3 \) into the formula:
\[ P(X = 4) = \binom{5}{4} (0.7)^4 (0.3)^{5-4} \]
\[ P(X = 4) = 5 \times (0.7)^4 \times (0.3)^1 \]

6. **Calculate \( (0.7)^4 \) and \( (0.3)^1 \)**:
- \( (0.7)^4 = 0.7 \times 0.7 \times 0.7 \times 0.7 = 0.2401 \)
- \( (0.3)^1 = 0.3 \)

7. **Combine the results**:
\[ P(X = 4) = 5 \times 0.2401 \times 0.3 \]
\[ P(X = 4) = 5 \times 0.07203 = 0.36015 \]

8. **Final answer**:
The probability that out of 5 persons, 4 are swimmers is approximately \( 0.36015 \).