To solve this problem, we will use the binomial probability formula:

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

where:

• $n = 30$ (the number of trials),
• $x = 5$ (the number of successes),
• $p = \frac{1}{5}$ (the probability of success on a single trial),
• $1-p = \frac{4}{5}$ (the probability of failure on a single trial),
• $\binom{n}{x}$ is the binomial coefficient, which can be calculated as $\frac{n!}{x!(n-x)!}$.

Step-by-Step Calculation

1. Binomial Coefficient: $\binom{30}{5} = \frac{30!}{5!(30-5)!} = \frac{30 \times 29 \times 28 \times 27 \times 26}{5 \times 4 \times 3 \times 2 \times 1} = 142506$

2. Calculating the Probability: We now substitute into the binomial formula: $P(X = 5) = 142506 \times \left(\frac{1}{5}\right)^5 \times \left(\frac{4}{5}\right)^{25}$

3. Simplify: $P(X = 5) = 142506 \times \frac{1}{3125} \times \left(0.8\right)^{25}$

4. Evaluate Powers: $\left(0.8\right)^{25} \approx 0.003174$

5. Final Probability Calculation: $P(X = 5) \approx 142506 \times \frac{1}{3125} \times 0.003174 \approx 0.067$

Answer:

The probability of exactly 5 successes is approximately 0.067.

So, the correct answer is:

D. 0.067

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Would you like to see more examples, or have any questions about this problem? Here are some related questions to expand your understanding:

1. How would the probability change if the number of trials were increased to 40?
2. What happens to the binomial probability when $p = 0.5$?
3. How do you interpret the binomial distribution in terms of a real-world scenario?
4. What is the relationship between the binomial distribution and the normal distribution for large $n$?
5. How would the calculation change if you were looking for the probability of at least 5 successes?

Tip: When working with binomial probabilities, always check if the problem can be approximated using a normal distribution for large $n$. This can save computation time.