Suppose a binomial probability experiment is conducted with $n=10$ trials and probability of success $p=0.3$. What is the probability of exactly $4$ successes?

• A. $120{0.3}^4{0.7}^6$
• B. $210{0.7}^4{0.3}^6$
• C. $10{0.3}^4{0.7}^6$
• D. $210{0.3}^4{0.7}^6$

Answer: (D)

Verified step by step guidance

1. Identify the parameters of the binomial experiment: number of trials \(n = 10\) and probability of success in each trial \(p = 0.3\).
2. Recognize that the problem asks for the probability of exactly 4 successes, so \(x = 4\).
3. Recall the binomial probability formula: \[P(X = x) = \binom{n}{x} p^{x} (1-p)^{n-x}\] where \(\binom{n}{x}\) is the binomial coefficient representing the number of ways to choose \(x\) successes out of \(n\) trials.
4. Calculate the binomial coefficient \(\binom{10}{4}\), which counts the number of ways to get exactly 4 successes in 10 trials.
5. Substitute the values into the formula: \[P(X=4) = \binom{10}{4} (0.3)^{4} (0.7)^{6}\] This expression gives the probability of exactly 4 successes.