This may be more of a mathematics question but it comes about in a physics context. I have a box of area $64cm \times 32cm$ and it is partitioned into two parts using a divider which can slide along one axis if hit by a ball.There are $N_{L}$ balls on the left side and $N_{R}$ on the right, all balls have radius $R=4$.
My question is can I find out the probability of my slider being in a particular interval.
Originally I partitioned the box into an $8 \times 4$ grid and calculated the number of $N$ balls into $n$ boxes. However I want a continuous model and am a bit confused about how exactly I could go about this. I feel like I will need to integrate of the area available for any particular ball from $x=8$ to $x=56$, where x is the position of the slider.
Any help is greatly appreciated!