Assuming I have 3 normally distributed random variables with different variance and mean:
$ X\sim N(\mu_1, \sigma_1^2) $, $ Y\sim N(\mu_2, \sigma_2^2) $ , $ Z\sim N(\mu_3, \sigma_3^2) $
According to this post, I know how to calculate the probability $P(X<Y)$.
Now I'm interested in the probability that X is smaller than the two other distributions and I calculate this as
$$ p_x = P((X<Y) \cap(X<Z)) = P(X<Y) \times P(X<Z) $$ by multiplying two $ erfc $ functions mentioned in the link. I can do the same by comparing if $Y$ is smaller than $Z$ and $X$. Eventually I get $p_y$ and $p_z$.
My questions:
1. Is that process correct?
2. If yes, I thought that $ p_x + p_y + p_z = 1$ but I get a lower probability when I try examples.