### 538 Riddler: Rebel vs. Stormtroopers

#### 26 December ’16

In this week's Riddler, we are rebels trying to defeat a group of 9 advancing stormtroopers. Fortunately for us, we are more accurate than the notoriously inaccurate stormtroopers, and the stormtroopers are clumped together, making them easy to pick off.

First, the hit / miss probabilities for the stormtroopers / rebel with N stormtroopers remaining:

Probability | Hit | Miss |
---|---|---|

Stormtrooper | ||

Rebel |

The probability that the rebel shoots one of the N remaining stormtroopers before being shot can be expressed as follows:

Putting this together for the entire battle, this is the equation we need to solve:

That denominator is very hairy (and the word "approximately" in the prompt makes me think closed form is a bit of a pipe dream), so I went at this empirically. Easy enough with Excel Goal Seek or Wolfram Alpha. The rebel and stormtroopers are evenly matched when . It gets a lot tougher for the rebel if the stormtroopers aren't clumped together; in that case, for the battle to be evenly matched.