3. A S1 bet on a single number on a casino's roulette wheel pays S$35 if the ball ends up in the number slot you choose. Here is the distribution of the payoff X: $35 0.026 Рayoff X Probability 0.974 Each spin of the roulette wheel is independent of other spins What are the mean and standard deviation of X? a. b. Sam comes to the casino weekly and bets on 10 spins of the roul ette wheel. What does the law of large numbers say about the average payoff Sam receives from his bets each visit? What does the central limit the or em say about the distribution of Sam's average payoff after betting on 520 spins in a year? c. d. Sam comes out ahead for the year if his average payoff is greater than S1 (the amount he bets on each spin). What is the probability that Sam ends the year ahead? The true probability is 0.396. Does using the central limit theorem provide a reasonable approximation?

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