Solved – Given a draw from one of two overlapping normal distributions, what is the probability it came from one vs. the other

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:

Two overlapping normal distributions

As Adrian already suggested you need to know the prior probability that $X$ came from each distribution. If $Y$ is an indicator telling us whether or not $X$ came from distribution one and $p_1$ and $p_2$ are the mixing (prior) probabilities then

$$ P(Y = 1 mid X = x) = frac{p_1 f_{X mid Y=1}(x)}{p_1 f_{X mid Y=1}(x) + p_2 f_{X mid Y=0}(x)} . $$

All you've specified are the conditional densities $f_{X mid Y = 1}$ and $f_{X mid Y = 0}$ but this isn't enough to calculate the probability. You also need to know something about $p_1$ and $p_2$.

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