Which statement is not true for a binomial distribution with $n=15$ and $p=\frac{1}{20}$?

• A. The distribution is symmetric.
• B. The variance is $0.7125$.
• C. The expected value is $0.75$.
• D. The probability of zero successes is greater than the probability of one success.

Answer: (A)

Verified step by step guidance

1. Identify the parameters of the binomial distribution: number of trials \(n = 15\) and probability of success in each trial \(p = \frac{1}{20} = 0.05\).
2. Calculate the expected value (mean) using the formula \(E(X) = n \times p\). Substitute the values to find \(E(X) = 15 \times 0.05\).
3. Calculate the variance using the formula \(Var(X) = n \times p \times (1 - p)\). Substitute the values to find \(Var(X) = 15 \times 0.05 \times (1 - 0.05)\).
4. Evaluate the symmetry of the binomial distribution. Recall that a binomial distribution is symmetric only when \(p = 0.5\). Since \(p = 0.05\), determine if the distribution is symmetric or skewed.
5. Compare the probabilities of zero successes and one success using the binomial probability formula: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\). Calculate \(P(X=0)\) and \(P(X=1)\) and compare their values.