A (solitaire) game starts with 3 ∈ N silver dollars in the pot. At each turn the number of silver dollars in the pot is counted (call it K) and the following procedure is repeated K times: a die is thrown, and according to the outcome the following four things can happen
If the outcome is 1 or 2 the player takes 1 silver dollar from the pot.
If the outcome is 3 nothing happens.
If the outcome is 4 the player puts 1 extra silver dollar in the pot (you can assume that the player has an unlimited supply of silver dollars).
If the outcome is 5 or 6, the player puts 2 extra silver dollars in the pot.
If there are no silver dollars on in the pot, the game stops.
Compute the expected number of silver dollars in the pot after turn n ∈ N.
Compute the probability that the game will stop eventually.
Let mn be the maximal possible number of silver dollars in the pot after the n-th turn? What is the probability that the actual number of silver dollars in the pot after n turns is equal to mn - 1?