Consider three i.i.d random variable $X_{0,1,2}\in [0,1]$ given by pdf $f(x)$ and cdf $F(x)$, then, We can calculate the probability, $$ P(X_0<X_1,X_2) = \int_{0}^1 f(x_0)\int_{x_0}^1f(x_1) \; dx_1\int_{x_0}^1f(x_2)d(x_2)\;dx_0 \\= \int_{0}^1 f(x_0)[1-F(x_0)]^2\;dx_0\\ =\frac{1}{3} $$ This is equal to 1/3 either by using probabilistic reasoning, i.e. the probability that 1 of these 3 variables is the smallest is 1/3 (as they all have the same chance of being the smallest) or by using the integral formula, $$ f(x)F^{n-1}(x)=\frac{1}{n}\frac{dF^n(x)}{dx} $$ and calculating the integral directly.
I am struggling to generalise this in the multivariate case, where $\bf{X_{0,1,2}}\in[0,1]^N$ are $N-$dimensional random variables from joint distribution $f(x_1,...,x_N)$ and cdf $F(x_1,...,x_N)$. Here we no longer have the integral formula nor the probabilistic intuition in order to calculate the expression, $$ P(\bigcap_{i=1}^N \{X_{0,i}<X_{1,i},X_{2,i}\}) $$
Any hints or solutions welcome.