2016 UNCO Math Contest II Problems/Problem 5

Problem

Rock and Roll

Zeus has decreed that Sisyphus must spend each day removing all the rocks in a certain valley and transferring them to Mount Olympus. Each night, each rock Sisyphus places on Mount Olympus is subject to the whims of Zeus: it will either be vaporized (with probability 10%), be rolled back down into the valley (with probability 50% ), or be split by a thunderbolt into two rocks that are both rolled down into the valley (with probability 40%). When the sun rises, Sisyphus returns to work, delivering rocks to Olympus. At sunrise on the first day of his punishment, there is only one rock in the valley and there are no rocks on Mount Olympus. What is the probability that there are exactly two rocks in the valley at sunrise on the third day? (If a rock is vaporized, it is gone.)

Solution