Final answer:
The student's question involves finding the optimal mixed strategy for a game using the minimax criterion and calculating the expected average winnings and advantage between the player and the house.
Step-by-step explanation:
The question is asking us to analyze a game from the perspective of player 1 (the evens player) and determine the optimal mixed strategy using the minimax criterion. To find the maximin point, we need to consider the worst possible outcomes for player 1 for each of his strategies and choose the strategy that provides the highest payoff among these minimum values. Once the maximin point is identified, player 1 can determine the probabilities to assign to each of his strategies to maximize his minimum guaranteed payoff, assuming that player 2 is playing optimally. This optimal mix of strategies that player 1 should play is known as the optimal mixed strategy.
As for the expected average winnings and whether the player or the house has an advantage, one would calculate the expected values for each possible outcome of the game, and compare these values to determine who, on balance, stands to benefit from repeated play. This could involve computing the long-term monetary return to the player per game and deciding whether the odds are favorable enough to make playing the game a rational choice based on the expected average payouts.