Shooting bullets This is from http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/May2014.html
Every second, a gun shoots a bullet in the same direction at a random constant speed between 0 and 1.
The speeds of the bullets are independent uniform random variables. Each bullet keeps the exact same speed and when two bullets collide, they are both annihilated.
After shooting $n$ bullets, prove that the probability that eventually all the bullets will be annihilated is zero if $n$ is odd and $\prod_{i=1}^{n/2} \frac{2i-1}{2i}$ when $n$ is even.
I tried to write recursion without success and Markov chain's but I don't see how them helps here. The case of $n\equiv 1 \pmod 2$ seems to be trivial.
 A: I'm wondering if we actually need any more information about those velocities.
Let's take the case in which $n=2.$ The formula says the probability of total annihilation should be $\frac38=\frac9{24}.$
The velocities are $v_1<v_2<v_3<v_4,$ and I'll refer to these velocities by their subscripts.
There are 24 possible orders in which the bullets are fired.
In 7 of those 24 orders, annihilation is certain: 2143, 3142, 3241, 3412, 4132, 4231, 4321.
In 13 of the 24, annihilation will not happen (which includes all 1xxx and xxx4 orders).
But that leaves the following four orders: 2431, 3421, 4213, 4312.
In each of those four, whether total annihilation occurs or not depends on which order the collisions take place. And that depends on what the velocities are.
Is there any way to partition down these four events so they split evenly? Maybe, but I haven't worked out the details.
For instance, if 31 happens before 43 then 2431 becomes "no" and 4312 becomes "yes", leaving the other two undecided. That sort of thing.
