Stacking boxes into a limited height box, all with same width and length, only heights change

Question

My problem is as follows:
I have ordered boxes to stack into a huge box.
All the boxes including the huge box have the same width and length.
The huge box has a constant height $H$.
All the other boxes have random heights $h_i$, that are of smaller or equal height as the huge box, $h_i \leq H$.
I am trying to find the probability distribution of having n boxes that fill into the huge box. That is:
$P(1~\text{box})= P(\ h_1 \leq H\ \& \ h_1+h_2 > H)$
$P(2~\text{boxes})= P(\ h_1+h_2 \leq H\ \&\ h_1+h_2+h_3 > H)$ and so on. These expressions are what I am trying to develop. Can we consider $P(h_1 \leq H)$ and $P(h_1+h_2 > H)$ independent? In this case, it is the simple multiplication of the probabilities. Otherwise, how can I go through calculations.

Answer 1

The space of outcomes of the $n$ heights is the hypercube with sides equal to $H$. Its volume is $H^n$.

The space of heights that will fit into the box is $$0\le x_1+...+x_n\le H$$ This forms a simplex in $\mathbb R^n$ whose volumes is $\frac{H^n}{n!}$

Dividing the two gives the probability as $\frac{1}{n!}$