Round #1041

A factory produces widgets and 2% are defective. If you sample 50 widgets, what is the probability that exactly 1 is defective?

```csharp using System; class Program { static void Main() { // define the probability of a widget being defective double p = 0.02; // define the number of widgets sampled int n = 50; // define the number of defective widgets we are interested in int k = 1; // calculate the probability using the binomial distribution double probability = BinomialProbability(n, k, p); Console.WriteLine("The probability that exactly 1 widget is defective: " + probability); } static double BinomialProbability(int n, int k, double p) { return Combination(n, k) * Math.Pow(p, k) * Math.Pow(1 - p, n - k); } static double Combination(int n, int k) { double result = 1; for (int i = 1; i <= k; i++) { result = result * (n - i + 1) / i; } return result; } } ```

```csharp using System; public class Program { public static void Main() { double probability = CalculateProbability(50, 0.02, 1); Console.WriteLine($"The probability that exactly 1 widget is defective is {probability:F4}"); } public static double CalculateProbability(int n, double p, int k) { // Calculate the probability using the binomial distribution formula double probability = BinomialCoefficient(n, k) * Math.Pow(p, k) * Math.Pow(1 - p, n - k); return probability; } public static double BinomialCoefficient(int n, int k) { // Calculate the binomial coefficient using the formula n! / (k! * (n-k)!) double result = 1; for (int i = 1; i <= k; i++) { result = result * (n - i + 1) / i; } return result; } } ``` This C# program calculates the probability of exactly 1 defective widget in a sample of 50, given a 2% defect rate. The `CalculateProbability` function uses the binomial distribution formula, which is `P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))`, where `n` is the sample size, `k` is the number of defective widgets, `p` is the probability of a widget being defective, and `nCk` is the binomial coefficient. The `BinomialCoefficient` function calculates the binomial coefficient using the formula `n! / (k! * (n-k)!)`. The result is then printed to the console.