A roulette payoff. A $1 bet on a single number on a casinos roulette wheel pays $35 if the ball ends up in the number slot you choose. Here is the distribution of the payoff X: Payoff X $0 $35 Probability 0.974 0.026 Each spin of the roulette wheel is independent of other spins. (a) What are the mean and standard deviation of X? (b) Sam comes to the casino weekly and bets on 10 spins of the roulette wheel. What does the law of large numbers say about the average payoff Sam receives from his bets each visit? (c) What does the central limit theorem say about the distribution of Sams average payoff after betting on 520 spins in a year? (d) Sam comes out ahead for the year if his average payoff is greater than $1 (the amount he bet 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? We will return to this problem in the next section.

Q:A roulette payoff. A $1 bet on a single number on a casinos roulette wheel pays $35 if theball ends up in the number slot you choose. Here is the distribution of the payoff X:(a) What are the mean and standard deviation of X(b) Sam comes to the casino weekly and bets on 10 spins of the roulette wheel. What doesthe law of large numbers say about the average payoff Sam receives from his bets eachvisit(c) What does the central limit theorem say about the distribution of Sams average payoffafter betting on 520 spins in a year (d) Sam comes out ahead for the year if his average payoff is greater than $1 (the amounthe bet on each spin). What is the probability that Sam ends the year ahead The trueprobability is 0.396. Does using the central limit theorem provide a reasonableapproximation We will return to this problem in the next section. Step by Step solution:Step 1 of 4:(a) What are the mean and standard deviation of XThus, mean and standard deviation of X is 0.91 and 5.569 respectively.