Probability that at least one of the bullets will go on forever. There's a gun located on an infinite line, let's say at 0 on number line.
It starts shooting bullets along that line, +X axis, at the rate of one bullet per second.
Each bullet has a velocity in the range [0, 1] m/s randomly chosen from a uniform distribution. 
If two bullets collide (are at the same spot at the same time) they explode and disappear. 
What is the probability that at least one of the infinite bullets will infinitely fly without colliding with another bullet?
 A: Let $X_n\stackrel{\mathrm{iid}}\sim U(0,1)$. For all positive integers $n$ and $j$ we have
$$\mathbb P\left(\bigcap_{k=n+1}^{n+j}\{X_k\leqslant X_n\} \right) = 2^{-j}, $$
and $$\bigcap_{k=n+1}^{n+j}\{X_k\leqslant X_n\}\supset \bigcap_{k=n+1}^{n+1+j}\{X_k\leqslant X_n\}, $$
and therefore
\begin{align}
\mathbb P\left(\bigcap_{k=n+1}^{\infty}\{X_k\leqslant X_n\} \right)  &= \mathbb P\left(\bigcap_{k=n+1}^{\infty}\{X_k\leqslant X_n\} \right)\\
&= \lim_{j\to\infty}P\left(\bigcap_{k=n+1}^{n+j}\{X_k\leqslant X_n\} \right)\\
&= \lim_{j\to\infty}2^{-j}\\
&=0.
\end{align}
Hence
$$
\mathbb P\left(\bigcup_{n=1}^\infty\bigcap_{k=n+1}^\infty\{X_k\leqslant X_n\} \right) \leqslant \sum_{n=1}^\infty \mathbb P\left(\bigcap_{k=n+1}^\infty\{X_k\leqslant X_n\} \right)=0,
$$
so that
$$\mathbb P\left(\bigcap_{n=1}^\infty\bigcup_{k=n+1}^\infty\{X_k> X_n\} \right) = 1 - \mathbb P\left(\bigcup_{n=1}^\infty\bigcap_{k=n+1}^\infty\{X_k\leqslant X_n\} \right)=1. $$
For each bullet $n$, there is a bullet $k>n$ traveling faster than bullet $n$ with probability $1$. It follows that the probability of at least one bullet flying forever without collision is zero.
