2016 UNCO Math Contest II Problems/Problem 6

Problem

Rock and Roll Forever?

(a) Given the situation in Question , what is the probability that Sisyphus must labor forever? That is, if Sisyphus begins with one rock in the valley on his first morning, what is the probability that the Olympian rocks are never all vaporized? (b) Suppose that the whims of Zeus obey the following rules instead: a rock will either be vaporized (with probability 10%), be rolled back down into the valley (with probability 20%), be split by a thunderbolt into two rocks that are both rolled down into the valley (with probability 30%), or be split by two thunderbolts into three rocks that are all rolled down into the valley (with probability 40%). Now what is the probability that Sisyphus must labor forever?

Solution

(a)  (b)