TITLE: Colliding Bullets
QUESTION [59 upvotes]: I saw this problem yesterday on reddit and I can't come up with a reasonable way to work it out.


Once per second, a bullet is fired starting from $x=0$ with a uniformly random speed in $[0,1]$. If two bullets collide, they both disappear. If we fire $N$ bullets, what is the probability that at least one bullet escapes to infinity? What if we fire an infinite number of bullets?

Attempt.

If $N$ is two, then it's equal to the probability that the first bullet is faster than the second, which is $\dfrac{1}{2}$.

If $N$ is odd, the probability of three bullets or more colliding in the same spot is $0$, so we can safely ignore this event. And since collisions destroy two bullets then there will be an odd number of bullets at the end. So at least one escapes to infinity.


For infinite bullets, I suspect that no single bullet will escape to infinity, but that they'll reach any arbitrarily big number. Although, I'm not sure on how I'd begin proving it.
Is there a closed form solution for even $N$? Or some sort of asymptotic behavior?

REPLY [2 votes]: Just for reference, I found a recent (yes, it's a today's posting!) arXiv paper about this problem and its variants.
https://arxiv.org/abs/1709.00789
If I skimmed over correctly, it considers various scenarios about randomness of bullet speeds and shooting interval and shows that the number of survival bullets have all the same distributions.