The law of large numbers tells us what happens in the long run.
Like many games of chance, the numbers racket has outcomes so variable - one three-digit number wins $600 and all others win nothing - that gamblers never reach "the long run."
Even after many bets, their average winnings may not be close to the mean.
For the numbers racket, the mean payout for single bets is $0.60 (60 cents) and the standard deviation of payouts is about $18.96.
If Joe plays 350 days a year for 40 years, he makes 14,000 bets.
(a)What is the mean of the average payout x that Joe receives from his 14,000 bets?
(b)What is the standard deviation of the average payout x that Joe receives from his n=14,000 bets?
(c)The central limit theorem says that his average payout is approximately Normal with the mean and standard deviation you found above.
What is the approximate probability that Joe's average payout per bet is between $0.50 and $0.70?
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