1. Suppose you have 11 boxes, numbered 1 to 11, and each box contains three balls, of which two are numbered 1 and one is numbered 2. A coin has a probability of 0.6 of coming up heads. You are going to toss the coin 11 times, and if a head occurs on the ith toss, you are going to take box number i and randomly select a ball. Let g denote the probability function of the random variable S, the total of all the selected balls. Given that g(10) = 0.1386 and g(11) = 0.1055, find g(12).
2. People with an initial wealth of 10 are offered a chance to play a game in which they win either 2 or 0, each with probability 1/2. If their utility function is given by the convex function u(x) = x2, what is the most they will pay to play this game.