Firstly, we focus on one box only:
For each thrown stone, the probability that a stones falls in this particular box is . The probability that it falls on another box is .
Figure 4: Probability tree leads to a binomial distribution.
If we throw stones, we can build a binary tree of height . In each tree's node, we choose the left path if the stone falls into the particular box we focus. The right path is chosen if it falls in another one. The probability that no stone falls in our box after throws is and follows the rightmost arm of the tree with height .
In general, the probability to have stones into one box follows the binomial distribution. With , we state
Figure 5: Dependence versus independence
Here, it is important to notice the following: while is the probability that one particular box is empty, it is not possible to generalize with to the problem where all boxes have at least one stone; the boxes are not independent from each other: a stone that does not land into a box does not fall apart but lands in another box, as in Figure 5.