For the binomial distribution with parameters $n$ and $p$, which of the following expressions gives the probability of exactly $k$ successes in $n$ independent trials?

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

Answer: (A)

Verified step by step guidance

1. Recall that the binomial distribution models the number of successes in \( n \) independent trials, each with success probability \( p \). The probability of exactly \( k \) successes is given by the binomial probability formula.
2. The formula involves three components: the number of ways to choose which \( k \) trials are successes, the probability of those \( k \) successes occurring, and the probability of the remaining \( n-k \) trials being failures.
3. The number of ways to choose \( k \) successes out of \( n \) trials is given by the binomial coefficient \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
4. The probability of exactly \( k \) successes is \( p^k \), since each success has probability \( p \) and the trials are independent.
5. The probability of the remaining \( n-k \) failures is \( (1-p)^{n-k} \), since each failure has probability \( 1-p \). Combining these, the probability of exactly \( k \) successes is: \[ \frac{n!}{k!(n-k)!} \times p^k \times (1-p)^{n-k} \]