(13 pts) Suppose that you repeatedly play a game where you roll a fair six-sided die. If the number of dots faces up is 6, you win $2; otherwise, you win S0. Let S, be the total amount that you win after n games. Assume S, = 0. (a) Find E[S4s], Var[S15), and C,(15, 20) = Cov(S15,Sz0). (b) Find the probability that you win $4 in 15 games, P[Sis 4]. (c) Find a mathematical expression for P[S20 = 8|S;5 = 4]. You do not need to evaluate it numerically. Just give an expression that could be evaluated numerically using Python or a calculator. (d) Find a mathematical expression for P[S20 =8n Sis = 4). (e) Now suppose that you start out with $5 (So = 5), but that cach time you play the game costs you $1. What is the probability that you still have money after playing the game 8 times (S, > 0)?

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