Suppose a binomial experiment consists of $n$ independent trials, each with probability of success $p$. Which of the following expressions gives the probability of observing exactly $k$ successes?

• A. $p^n\cdot {(1-p)}^k$
• B. $\frac{n!}{k!(n-k)!}\cdot p^n\cdot {(1-p)}^k$
• C. $\frac{n!}{k!}\cdot p^k$
• D. $\frac{n!}{k!(n-k)!}\cdot p^k\cdot {(1-p)}^{n-k}$

Answer: (D)

Verified step by step guidance

1. Understand that a binomial experiment consists of \( n \) independent trials, each with a probability of success \( p \), and we want the probability of exactly \( k \) successes.
2. Recall the binomial probability formula: the probability of exactly \( k \) successes in \( n \) trials is given by the number of ways to choose \( k \) successes multiplied by the probability of those successes and the probability of the remaining failures.
3. Express the number of ways to choose \( k \) successes from \( n \) trials using the binomial coefficient: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
4. Write the probability of \( k \) successes as \( p^k \) and the probability of \( n-k \) failures as \( (1-p)^{n-k} \).
5. Combine these components to form the binomial probability formula: \[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} = \frac{n!}{k!(n-k)!} \cdot p^k \cdot (1-p)^{n-k} \].