Probability of $n$ uniformly distributed points occuring within $m$ units of distance of each other

Question

Suppose $U_i \sim \mathcal{U_{[0, 1]}}$ are independent uniformly distributed random variables over the interval $[0, 1]$. I believe the notation for the probability of two realisations of $U_i$ occurring within 0.5 units of each other is the following: $$ \mathbb{P} \Bigl( \lvert U_1 - U_2 \rvert < \frac{1}{2} \Bigr) . $$ To generalise the statement above, am I correct to say that the probability of $n$ realisations of $U_i$ occurring within $0 \leq m \leq 1$ units of each other is $$ \Bigl[ \mathbb{P} \Bigl( \lvert U_1 - U_2 \rvert < m \Bigr) \Bigr]^p $$ where $p = {n \choose 2}$?

Answer 1

By "$n$ realisations of $U_i$ occurring within $0 \leq m \leq 1$" you mean that all pairs are at distance less than $m$ ? That would be $P(\max_{i,j} | U_i - U_j| < m)$ or , equivalently $$P( U_{(n)} - U_{(1)} < m)$$ where $U_{(n)}$,$U_{(1)}$ are resp the maximum and the minimum.

Of course, the above only amounts to denoting the event, not to computing its probability.

For that, see here.