Fighting stormtroopers

This Riddler puzzle is about fighting a group of stormtroopers. Why are they so inaccurate anyway?

In Star Wars battles, sometimes a severely outnumbered force emerges victorious through superior skill. You panic when you see a group of nine stormtroopers coming at you from very far away in the distance. Fortunately, they are notoriously inaccurate with their blaster fire, with only a 0.1 percent chance of hitting you with each of their shots. You and each stormtrooper fire blasters at the same rate, but you are $K$ times as likely to hit your target with each shot. Furthermore, the stormtroopers walk in a tight formation, and can therefore create a larger target area. Specifically, if there are $N$ stormtroopers left, your chance of hitting one of them is $(K\sqrt{N})/1000$. Each shot has an independent probability of hitting and immediately taking out its target. For approximately what value of $K$ is this a fair match between you and the stormtroopers (where you have 50 percent chance of blasting all of them)?

Here is my solution.
[Show Solution]

3 thoughts on “Fighting stormtroopers”

  1. Maybe slightly simpler (but still numerical): Facing N stormtroopers, your chance of killing a stormtrooper before they kill you is (p-pq)/(p + q – pq), where p is your chance of killing with each shot, that is, (K*sqrt(N))/1000, and q is their chance of killing you with one shot each, which is one minus the chance of their all missing, or (1 – (.999)^N). Your chance of survival is the product of the chances of your killing first for N from 9 to 1.

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