I have a stochastic event with two possible outcomes ($A$ and $B$). I also have two measurement techniques, a direct and an indirect one. The direct one can tell me exactly the outcome, but it implies interrupting the process.
The indirect one measures an intermediate variable $x$, which also has some variation. From previous experiments, I know that if the outcome is $A$, then the measured values of $x$ would present a normal probability distribution with mean $\bar{x}_A$ and standard deviation $\sigma_A$. Similarly, if the outcome is $B$, then the measured values of $x$ would present a normal probability distribution with mean $\bar{x}_B$ and standard deviation $\sigma_B$.
Given this information and a new measurement $\hat{x}$ form an unknown process outcome (which I know must be $A$ or $B$), I need to compute what is the probability of the outcome being $A$ or $B$.
I saw this other post with a similar question, but the answer seems overly complicated since I need to implement this in a computer for online classification of the samples.