The probability that a student is not a swimmer is 1/5. Then find the probability that out of 5 students exactly 4 are swimmer.

Explanation

To solve the problem, we need to find the probability that out of 5 students, exactly 4 are swimmers, given that the probability that a student is not a swimmer is \( \frac{1}{5} \).

### Step-by-Step Solution:

1. **Identify the probabilities:**
- The probability that a student is not a swimmer (denote as \( P(S') \)) is given as \( \frac{1}{5} \).
- Therefore, the probability that a student is a swimmer (denote as \( P(S) \)) is:
\[ P(S) = 1 - P(S') = 1 - \frac{1}{5} = \frac{4}{5} \]

2. **Set up the parameters for the binomial probability formula:**
- We are dealing with a binomial distribution since we have a fixed number of trials (students), two possible outcomes (swimmer or not swimmer), and a constant probability of success.
- Let \( n = 5 \) (the total number of students).
- Let \( r = 4 \) (the number of students who are swimmers).
- Let \( p = P(S) = \frac{4}{5} \) (the probability of success).
- Let \( q = P(S') = \frac{1}{5} \) (the probability of failure).

3. **Use the binomial probability formula:**
The formula for the probability of getting exactly \( r \) successes in \( n \) trials is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
Substituting the values:
\[ P(X = 4) = \binom{5}{4} \left(\frac{4}{5}\right)^4 \left(\frac{1}{5}\right)^{5-4} \]

4. **Calculate \( \binom{5}{4} \):**
\[ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \cdot 1!} = \frac{5 \cdot 4!}{4! \cdot 1} = 5 \]

5. **Calculate \( \left(\frac{4}{5}\right)^4 \):**
\[ \left(\frac{4}{5}\right)^4 = \frac{4^4}{5^4} = \frac{256}{625} \]

6. **Calculate \( \left(\frac{1}{5}\right)^{5-4} \):**
\[ \left(\frac{1}{5}\right)^{1} = \frac{1}{5} \]

7. **Combine all parts:**
\[ P(X = 4) = 5 \cdot \frac{256}{625} \cdot \frac{1}{5} \]
The \( 5 \) in the numerator and denominator cancels out:
\[ P(X = 4) = \frac{256}{625} \]

8. **Final calculation:**
To convert \( \frac{256}{625} \) to decimal form:
\[ \frac{256}{625} = 0.4096 \]

### Conclusion:
The probability that out of 5 students, exactly 4 are swimmers is \( 0.4096 \).