## Drawing from i.i.d. distribution. An analytical solution?

Given $N$ i.i.d. random variables $X_1, X_2, ... X_N$ with distribution $N(0, \sigma)$

Question 1: What is the probability that there exists a set $A$ of the $X_is$ such that 1. $|A|=T$; 2. $|X_i - X_j| < w$ for all $X_i, X_j \in A$ and some real $w$.

Question 2: Let $B$ be the set of all such $A$s from Question 1, and define $A_{min} \in B$ such that $\max A_{min} < \max A_j$ for all $A_j \in B , A_j \neq A_{min}$. What's the probability that $\max A_{min} = r$ for some real number r.