What is the probability that $\max(X_1, X_2, \ldots, X_n) < \min(Y_1, Y_2, \ldots, Y_n)$ where the $X_i$'s and the $Y_i$'s $\sim U[0, 1]$?

Question

What is the probability that $\max(X_1, X_2, \ldots, X_n) < \min(Y_1, Y_2, \ldots, Y_n)$ where:

• $X_i \sim U[0, 1]$ and $Y_i \sim U[0, 1]$, and
• $X$ and $Y$ are independent continuous random variables?

I went as far as figuring out that the max and the min are beta-distributed, but I am not exactly sure how to go from there, and I am not even exactly sure whether this is the way to go. Any help is appreciated. Thanks!

Answer 1

Let $M=\max X_i$ and $m=\min Y_i.$ Then $\Pr(m>y)=(1-y)^n ,\Pr(M<x) =x^n, $ $$\Pr(M<m)= E(\Pr(M<m|m))=E(m^n)=n\int_0^1y^n(1-y)^{n-1}dy=nB(n+1,n)=\frac{n!^2}{(2n)!}.$$