Suppose that $P(x \, | \, \omega_i) \sim \mathcal{N}(\mu_i, \sigma^2)$ for $i = 1, 2$ and that $P(\omega_1) = P(\omega_2) = 1/2$. I'm trying to find a value $x$ such that $P(\omega_1 \, | \, x) = P(\omega_2 \, | \, x) = 1/2$. We assume that $\mu_2 \geq \mu_1$.
My intuition is telling me that since both random variables have the same variance, the value $x$ that I'm seeking is simply $(\mu_2 + \mu_1)/2$. However, I'm having a hard time showing this.
Could anyone point me in the right direction?