Throw darts at a wall

Question

You are throwing darts at a wall. There are two identical circular targets on the wall, each with radius $r$. Their centers lie on the same horizontal line and are separated by distance $d$.

You may choose an aiming point $x$ along the line connecting the two centers. However, due to random error, the actual landing point is not exactly $x$. Assume the horizontal landing position is normally distributed:

$$
Y \sim \mathcal{N}(x, \sigma^2)
$$

For simplicity, assume the targets are small relative to $\sigma$, so the probability of hitting a target is approximately proportional to the Gaussian density at that target’s center.

## Questions

1. Write an expression, up to a proportionality constant, for the probability of hitting either target as a function of $x$.

2. Determine when it is optimal to aim at the midpoint between the two targets.

3. Determine when it is better to aim closer to one of the targets.

4. Give an intuitive explanation of how the optimal strategy changes as $d/\sigma$ changes.

Accepted Answer

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