Suppose you have a role playing game in which n players attack a boss with H hit points. Each hit reduces the boss's hit points by a certain amount. The players take turns hitting, starting with player 1, proceeding to player 2, onward to player n, and then back to player 1 again. Eventually, some player deals the final blow and the boss dies. The question is: what's the probability a given player will deal this final blow? We want to look at the limit as the boss's hit points increase to infinity, to smooth out any irregularities. We'd expect that the probability a player deals the final blow is in proportion to the average damage he does.(adsbygoogle = window.adsbygoogle || []).push({});

To specify the problem, let X be a random variable whose domain is subsets of the positive integers. Each particular subset x indicates which points of damage player 1 dealt; for instance, if x contains {5, 6, 8} but not 7, then that means the fifth, sixth, and eighth points of damage dealt to the boss came from player 1, but the seventh point came from some other player. Let the boss's "base" hit points be a probability distribution H over the positive integers, which we will multiply by a constant c and take the limit as c increases.

I believe the following, which corresponds to the idea that the probability a player deals the final blow is in proportion to the average damage he does.

[tex]

\lim_{c \rightarrow \infty} P(cH \in X) = \lim_{k \rightarrow \infty} \mathbb{E}(\sum_{i=1}^{k} | \{ z \in X : z \leq i \} |) / k

[/tex]

for all distributions H and X, whenever both limits exist.

I don't have a proof of this however.

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# Probability of dealing the final blow

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