Question

Find \( P(X=2) \) if \( X \) is a binomial random variable with \( n=5 \) and \( p=0.3 \). Round your answer to three decimal places. \[ P(X=2)= \]

Answer 1

P(X=2) for a binomial random variable X with n=5 and p=0.3 is approximately 0.308.

Step-by-step explanation:

In a binomial distribution, P(X=k) is calculated using the probability mass function (PMF) formula: P(X=k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success on a single trial.

In this case, P(X=2) is calculated as P(X=2) = 5C2 * 0.3^2 * (1-0.3)^(5-2).

Using the binomial coefficient formula nCk = n! / (k!(n-k)!), where n! denotes the factorial of n, we find 5C2 = 5! / (2!(5-2)!) = 10.

Substituting this into the original formula, we get P(X=2) = 10 * 0.3^2 * (0.7)^3. Calculating this expression yields the final answer of approximately 0.308, rounded to three decimal places.

Therefore, the probability of obtaining exactly 2 successes in 5 trials for a binomial random variable with p=0.3 is approximately 0.308.