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In a room stand n armed and angry people. At each chime of a clock, everyone simultaneously spins around and shoots a random other person. The persons shot fall dead and the survivors spin and shoot again at the next chime. Eventually, either everyone is dead or there is a single survivor. As n grows, what is the limiting probability that there will be a survivor?

I have been trying to solve this problem for a while but couldn't find a good approach. Any insight is appreciated! Here are the probabilites for the small values of $n$:

$n = 2$ -> $0$

$n = 3$ -> $3 / 4$

$n = 4$ -> $48/81$

For much larger values of n, I ran some computer simulation and found that the answer gets close to $1 / 2$.

Transcendental
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jul 16 '23 at 06:47
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    I would recommend writing this problem down using less of a story, and more mathematics. – David Raveh Jul 16 '23 at 07:10
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    For $n = 2$, clearly the answer is 0. For ${n = 3}$, answer turns out to be ${3/4}$. For ${n = 4}$, the answer is ${48/81}$. For much larger $n$ I ran some simulations and found that the answer converges to $1/2$.

    I tried some recursive reasoning, but it quickly became intractable.

     – Transcendental Jul 17 '23 at 00:04
  • Thanks for the context! You should put that into the original post, since comments are prone to deletion. – Brian Tung Jul 18 '23 at 06:50
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    https://math.stackexchange.com/questions/2430193/group-russian-roulette?rq=1 – Manlio Jul 18 '23 at 06:54
  • Thanks @Manlio, that looks like the exact same problem :D – Transcendental Jul 18 '23 at 07:01
  • It doesn't converge to $1/2$. Since the behavior for large $n$ is that about $n/e$ survivors remain after a single round, the probability is actually oscillatory, repeating about every factor of $e$. Starting with $n=3$, my simulations with $10^6$ trials each give these probabilities: $[0.750553, 0.591565, 0.468419, 0.416254, 0.438881, 0.488488, 0.532949, 0.554202, 0.556286, 0.544488, 0.525525, 0.504147, 0.486463, 0.472484, 0.464400, 0.462105, 0.463823, \ldots]$. It's oscillating between $0.54$ and $0.46$ or so. – mjqxxxx Jul 22 '23 at 18:48
  • Here's the paper linked in the other question. – Varun Vejalla Jul 25 '23 at 02:17

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