Let \(X_1, X_2, \ldots, X_n\) be \(n\) mutually independent random variables, each of which is uniformly distributed on the integers from 1 to \(k\). Let \(Y\) denote the minimum of the \(X_i\)’s. We want to find the distribution of \(Y\).
\[ P(Y = y) = \left(\frac{y}{k}\right)^n - \left(\frac{y-1}{k}\right)^n \]
for \(y = 1, 2, \ldots, k\).