You toss a fair tetrahedral (4-sided, and the sides are marked "1", "2", "3", and "4") die, and you play the following game repeatedly. For each game, you throw the die n > 3 times. Starting from the third throw of the current game, right after each throw, you will get 1 dollar if the total number of points on the last three throws is 8 or higher, and you pay 1 dollar otherwise. It is possible to verify that your probability of winning one dollar on the third throw is 0.5. The random variable Xi represents the outcome of the i th throw. Required:
Find the expected value of your total gains/losses over one game in terms of n.