You drop 10 infinitely small m&m’s in the plane and they all land in a random place. Is it possible to cover all of these m&m’s with disks of radius 1, such that the disks do not overlap?
One can arrange the disk in a hexagonal circle packing on the plane. For a fixed configuration of m&m’s, this pattern of disks can now be placed in a random way on the plane. The fraction of the plane that is covered by the disks is \(\frac{\pi \sqrt{3}}{6} \approx 0.907\). Therefore the probability that an arbitrary m&m is covered by one of the disks is also approximately \(0.907\). By linearity of expectation, the expected total number of m&m covered by the disks following this procedure is approximately \(9.07\). Since this is an average, this implies that there has to be at least one way of placing the circle packing, such that the number of covered m&m’s is greater or equal to \(9.07\). Since every outcome of the number of covered m&m’s is an integer, we conclude that there exists a placement for which all m&m’s are covered.