Shoot me!!

Source: P. Winkler

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 probabality that there will be a survivor. :)

Treat at H8 canteen for the person solving it first :)
Solution: Posted by me in comments (the limiting probability does not exist, per Winkler)!

Comments

  1. A Rustle8:03 PM, October 12, 2009

    Is the answer 0?

    ReplyDelete
  2. Pratik Poddar9:53 PM, October 12, 2009

    nope :(

    ReplyDelete
  3. Pratik Poddar11:43 PM, October 14, 2009

    Since no one has been able to solve it till now, probably this would help.

    The limiting probability does not exist in the sense that the probability does not approach a unique value

    Since source is Winkler, you are sure that its correct :P

    ReplyDelete
  4. SB1236:03 AM, January 04, 2012

    Isn't it just e^-1 ??

    Because prob. of not being shot by a person when theyre are n people alive is 1-1/n

    Now prob. of at least one person alive we use inclusion exclsion principle, but when n is large this approx holds:

    Prob of not being shot is (1-1/n)^n after n round.

    So I get e^-1

    ReplyDelete
  5. Pratik Poddar12:52 PM, September 08, 2026

    That Poisson intuition is seductive, but it is not e^-1. The rounds are not independent of each other - who dies in one round changes the number of shooters in the next, and with an even number of survivors it is possible for everyone to die simultaneously. The real answer is stranger: the probability that anyone survives does not converge to a limit as n grows. It oscillates (slowly, as a function of log n). Winkler mentions this in his puzzle collection - a great example of a problem where the naive limiting argument completely fails. (replied using AI)

    ReplyDelete

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