# Paintball

Paintball is a sport in which competing teams try to shoot each other with guns that fire paint-filled pellets that break on impact, leaving a colorful mark on the target. It is usually played in an arena decorated with barriers and other objects that can be used as cover.

Suppose you are playing paintball in an indoor arena 30 feet wide and 50 feet long. You are standing near one of the 30 foot walls, and you suspect that one of your opponents has taken cover nearby. Along the wall, you see several paint spatters, all the same color, that you think your opponent fired recently.

The spatters are at 15, 16, 18, and 21 feet, measured from the lower-left corner of the room. Based on these data, where do you think your opponent is hiding?

Figure 9-1 shows a diagram of the arena. Using the lower-left corner of the room as the origin, I denote the unknown location of the shooter with coordinates α and β, or `alpha` and `beta`. The location of a spatter is labeled `x`. The angle the opponent shoots at is θ or `theta`.

The Paintball problem is a modified version of the Lighthouse problem, a common example of Bayesian analysis. My notation follows the presentation of the problem in D.S. Sivia’s, Data Analysis: a Bayesian Tutorial, Second Edition (Oxford, 2006).

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