For each $i\in \{ 1,\ldots, 10\},\ X_i\sim N(0,1);\ $ these random variables are independent. What is the probability that (at least) two of the $X_i$ are within $0.01$ of each other? And yes, this is equal to $1 - p($ all $ X_i $ are greater than $ 0.01$ apart $),$ but I don't know what to do with this.
If it were two observations rather than ten, you could just consider the distribution of $X_1-X_2.$ But with ten observations this becomes more complicated. Whilst I'm sure there is some convoluted way using combinatorics/inclusion/exclusion and sums, I am more interested in if there is some good approximation or some distribution I don't know about that will help.