In a binomial experiment with $4$ trials and probability of success $p=0.4$, what is the probability of obtaining exactly $3$ successes?

• A. $0.4096$
• B. $0.2304$
• C. $0.3456$
• D. $0.1536$

Answer: (D)

Verified step by step guidance

1. Identify the parameters of the binomial experiment: number of trials \(n = 4\) and probability of success in each trial \(p = 0.4\).
2. Recall the binomial probability formula for exactly \(k\) successes in \(n\) trials: \[P(X = k) = \binom{n}{k} p^{k} (1 - p)^{n - k}\]
3. Substitute the given values into the formula with \(k = 3\): \[P(X = 3) = \binom{4}{3} (0.4)^{3} (1 - 0.4)^{4 - 3}\]
4. Calculate the binomial coefficient \(\binom{4}{3}\), which represents the number of ways to choose 3 successes out of 4 trials.
5. Evaluate the powers of \(p\) and \$1-p$, then multiply all parts together to find the probability of exactly 3 successes.