Question: A lottery payoff. A $1 bet in a state lottery's Pick 3 game pays $500 if the three-digit number you choose exactly matches the winning number, which is drawn at random. Here is the distribution of the payoff \(X\) :

Each day's drawing is independent of other drawings.

1. What are the mean and standard deviation of \(X\) ?
2. Joe buys a Pick 3 ticket twice a week. What does the law of large numbers say about the average payoff Joe receives from his bets?
3. What does the central limit theorem say about the distribution of Joe's average payoff after 104 bets in a year?
4. Joe comes out ahead for the year if his average payoff is greater than $1 (the amount he spent each day on a ticket). What is the probability that Joe ends the year ahead?

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