An interesting stochastic game (Part I)

We consider the following game played by two players (P1, P2). Both players choose a bitstring of length n (e.g. n=3, then available bitstrings would be 010, 110, 001, 111, and so on). Now a 0/1-coin (i.e. a coin with one side 0 and the other side 1, both occur equally likely) is thrown and the string is recorded. The game ends as soon as the last n cointosses form a string belonging to one of the two players. The player whose string is hit, wins.

An amazon interview question (not completely solved)

Taken from this site:

Given n red balls and m blue balls and some containers, how would you distribute those balls among the containers such that the probability of picking a red ball is maximized, assuming that the user randomly chooses a container and then randomly picks a ball from that?

Perfect concentration game

Probably everyone knows the game concentration. For those who don't, here's an explanation. This article presents a stochastic problem concerning this game.