I'm developing some game theory models that involve normal distributions, and am not sure how to solve this probability problem:
Suppose there are two normal distributions: $$ X_1 \sim N(\mu_1, \sigma^2) $$ $$ X_2 \sim N(\mu_2, \sigma^2) $$
You know what both distributions are, but that's all you know. I take a draw (call it $s$) from one of the two distributions and show it to you. Given $s$, what is the probability I chose from $X_1$?
Thanks for any help!
Here's of a picture of two overlapping normal distributions in case it's helpful to have a visual: