Normal distribution comparing 3 or more variables Let's say we had 3 students about to run race and their previous run times each follow a normal distribution,

*

*$X \sim N(10, 2)$

*$Y \sim N(14, 3)$

*$Z \sim N(5, 1)$
Where $X$, $Y$ and $Z$ are independent.
I want to know if it is possible to work out the probability that $X$ will run the fastest time?
I feel like the question boils down to $P((X>Y) \cap (X>Z))$. I can work out $P(X>Y)$ and $P(X>Z)$  by creating a new distribution for each and just calculating mean and variance. But I am struggling to go this extra step to determine the probability that both scenarios are true.

If $\phi_1,\phi_2,\phi_3$ are the density function sof $X,Y$ and $Z$ then $$P((X>Y)\cap (X>Z))$$ $$=\int_{-\infty}^{\infty} \int_z^{\infty}\int_y^{\infty} \phi_1(x)\phi_2(y)\phi_3(z) dx dydz$$ $$+ \int_{-\infty}^{\infty}  \int_{-\infty}^{z}\int_z^{\infty} \phi_1(x)\phi_2(y)\phi_3(z) dx dydz$$
