MOTIVATION: I will quote Wikipedia's article on a soccer goalkeeper for the motivation:
Some goalkeepers have even scored goals. This most commonly occurs where a goalkeeper has rushed up to the opposite end of the pitch to give his team an attacking advantage in numbers. This rush is risky, as it leaves the goalkeeper's goal undefended. As such, it is normally only done late in a game at set-pieces where the consequences of scoring far outweigh those of conceding a further goal, such as for a team trailing in a knock-out tournament.
The mathematical question: Consider the following game (simplified soccer):
A single player starts with a score of 0.5 and plays N turns. In each turn, the player has to choose one of 2 strategies: $(p_{-1},p_0,p_1)$ or $(q_{-1},q_0,q_1)$ (these are probability vectors) and then her score is increased by -1, 0 or 1 according to the probabilites dictated by the chosen strategy. The player wins if at the end of the game she has a positive score, and loses if she has a negative score (the player's objective is to win, the only thing that matters is whether the final score is positive or negative).
What is the optimal global strategy given $N$, $(p_{-1},p_0,p_1)$ and $(q_{-1},q_0,q_1)$?
A global strategy is a function of the number of turns left, the current score and the 2 probability vectors (which are constant for all turns).
If this question is hard, it may still may interesting to approximate an optimal global strategy (in what sense?).