Explanation

The inverse cumulative distribution function (also called the quantile function) for a binomial distribution gives the smallest number of successes k such that P(X ≤ k) ≥ p, where p is the given probability.

For this problem:

• n = 30 trials
• p = 0.05 (probability of success)
• We want the inverse CDF for probability = 25.0% = 0.25

Let's calculate the cumulative probabilities:

• P(X = 0) = (1-0.05)^30 = (0.95)^30 ≈ 0.2146
• P(X ≤ 0) = 0.2146
• P(X = 1) = 30 × 0.05 × (0.95)^29 ≈ 30 × 0.05 × 0.2259 ≈ 0.3389
• P(X ≤ 1) = 0.2146 + 0.3389 = 0.5535
• P(X = 2) = C(30,2) × (0.05)^2 × (0.95)^28 ≈ 435 × 0.0025 × 0.2378 ≈ 0.2586
• P(X ≤ 2) = 0.5535 + 0.2586 = 0.8121

Now, looking for the smallest k where P(X ≤ k) ≥ 0.25:

• P(X ≤ 0) = 0.2146 < 0.25
• P(X ≤ 1) = 0.5535 ≥ 0.25

Therefore, the inverse CDF for probability 0.25 is 1 success, which corresponds to Option B.