In one lottery option in Canada, you bet on a six-digit number between 000000 and 999999. For a $1 bet, you win $100,000 if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are μ = 0.10 (that is, 10 cents) and σ = 100.00. Joe figures that if he plays enough times every day, eventually he will strike it rich, by the law of large numbers. Over the course of several years, he plays 1 million times. Let denote his average winnings.
a. Find the mean and standard deviation of the sampling distribution of .
b. About how likely is it that Joe’s average winnings exceed $1, the amount he paid to play each time? Use the central limit theorem to find an approximate answer.