Tiny Target II

You are shooting two targets on the real line, say $T_1$ and $T_2$ . The two targets are of radius $a\varepsilon$ and $b\varepsilon$ , both $<<< 1$ and are centered at $x_1 \neq x_2$ , respectively. This means $T_1$ covers $[x_1 - a\varepsilon, x_1 + a\varepsilon]$ , and similar for $T_2$ . Assume that the two targets are disjoint. You have one dart to shoot at the line. Your goal is to maximize your probability of hitting one of the targets. You can choose where to center your throw on the line. If you choose to center your dart at $\mu$ , the actual position that the dart lands on the real line is $X \sim N(\mu,\sigma^2)$ , where $\sigma^2$ is fixed. Find the value of $\mu$ that maximizes the probability of hitting a target when $x_1 = -4, x_2 = 0, a = 5, b = 2,$ and $\sigma^2 = 9$ . If necessary, round your answer to the nearest thousandth.

Answer format: round to 3 decimal places.