I'm trying to deduce the integral formula of the total error for the Bayes Optimal Classifier, in the case we have only two classes.
In the example i'm considering, after i've applied the bayes rule i got that the classification's border is at 4. As you can see in the picture.
The error of each classification is alread written in the picture. What i've tried to deduce is the total error, and to me :
$\varepsilon = P(\omega _{1}|X)\epsilon _{2}+P(\omega _{2}|X)\epsilon _{1}$
where X is the feature vector and $\omega _{1}$,$\omega _{2}$ are the 2 classes.
However, standing to my prof's slides, the correct formula is :
But i dont understand why :S