When the Midpoint Stops Winning

Two tiny targets of equal radius $\varepsilon$ sit on the real line, centred 5 units apart. You throw one dart aimed at $\mu$, and it lands at $X \sim N(\mu, \sigma^2)$. For small $\varepsilon$ the chance of hitting a target is proportional to the sum of the Normal density evaluated at the two centres.

Aiming at the midpoint is optimal for large $\sigma$ and is not optimal for small $\sigma$. Find the critical value $\sigma^{*}$ below which the midpoint stops maximising the hit probability.

Answer format: round to 2 decimal places.