This question already has an answer here:
Suppose you know $Y \sim N(\mu_1, \sigma_1^2)$ or $Y \sim N(\mu_2, \sigma_2^2)$. You observe $Y=y$, some realization of the random variable $Y$. What is the probability that $Y \sim N(\mu_1, \sigma_1^2)$?
My intuition is to compare $p$-values from each distribution. Let $p_i$ be the $p$-value for $y$ under $N(\mu_i, \sigma_i^2)$. Here I am thinking of the two-sided $p$-value, $p_i = 2\Phi(-|y-\mu_i|/\sigma_i)$ where $\Phi(x)$ is the standard normal distribution function. I would answer my own question as $p_1/(p_1+p_2)$. But I cannot find any reference that would support this (or even treats this problem).