20.8The numbers racket. Pick 3 lotteries (Example 1) copy the numbers racket, an illegal gambling operation common in the poorer areas of large cities. States usually justify their lotteries by donating a portion of the proceeds to education. One version of a numbers racket operation works as follows. You choose any one of the 1000 threedigit numbers 000 to 999 and pay your local numbers runner $1 to enter your bet.Each day, one three-digit number is chosen at random and pays off $600. What is the expected value of a bet on the numbers? Is the numbers racket more or less favorable to gamblers than the Pick 3 game in Example 1? Example 1The DC LotteryHere is a simple lottery wager: the "Straight" from the DC-3game of the DC Lottery offered by Washington, DC. You pay$1.00 and choose a three-digit number. The lottery chooses athree-digit winning number at random and pays you $500 ifyour number is chosen. Because there are 1000 three-digitnumbers, you have probability 1-in-1000 of winning. Here is theprobability model for your winnings:Outcome:$0$500Probability:0.9990.001What are your average winnings? The ordinary average of thetwo possible outcomes $0 and $500 is $250, but that makes nosense as the average winnings because $500 is much less likelythan $0. In the long run, you win $500 once in every 1000 betsand $0 on the remaining 999 of 1000 bets. (Of course, if you playthe game regularly, buying one ticket each time you play, afteryou have bought exactly 1000 DC-3 tickets, there is no guaranteethat you will win exactly once. Probabilities are only long-runproportions.) Your long-run average winnings from a ticket are1$500-+$0-1000999=$0.501000or 50 cents. You see that in the long run the state pays out one-half of the money bet and keeps the other half. The expected value of a random phenomenon that has numerical outcomes is found bymultiplying each outcome by its probability and then adding all the products.In symbols, if the possible outcomes are a1, a2,... , a; and their probabilities are pı, P2,... , Pk, the expected value is expected value = aip1 + a2P2 + ...+ a,pkHere is a general definition of the kind of "average outcome" weused to evaluate the bets in Example 1.An expected value is an average of the possible outcomes, but it isnot an ordinary average in which all outcomes get the same weight.Instead, each outcome is weighted by its probability so thatoutcomes that occur more often get higher weights.

Name: 20.8The numbers racket. Pick 3 lotteries (Example 1) copy the numbers
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Description: 20.8The numbers racket. Pick 3 lotteries (Example 1) copy the numbers racket, an illegal gambling operation common in the poorer areas of large cities. States usually justify their lotteries by donating a portion of the proceeds to education. One ver