Probabilities of detecting gamma or cosmic rays by HAWC with the free tiling game

The Buffon free tiling game is associated with the probability that the HAWC (High Altitude Water Cherenkov) observatory detects cascades created by high-energy cosmic rays and gamma rays. For this, all possible cases of the free tiling game are analyzed. With the association performed, a simple model is obtained to calculate the probabilities of cascades falling on HAWC and the outrigger ring in three possible cases. Each case is studied assuming the region occupied by the observatory as square or hexagonal. It is concluded that when considering hexagons the probabilities are slightly higher, but it may be more effective to consider squares. The calculations require only basic geometry.