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The 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?

The 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?

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
Problem 61E: Roulette American roulette is a game in which a wheel turns on a spindle and is divided into 38...
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The 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 1 The DC Lottery Here is a simple lottery wager: the "Straight" from the DC-3 game of the DC Lottery offered by Washington, DC. You pay $1.00 and choose a three-digit number. The lottery chooses a three-digit winning number at random and pays you $500 if your number is chosen. Because there are 1000 three-digit numbers, you have probability 1-in-1000 of winning. Here is the probability model for your winnings: Outcome: $0 $500 Probability: 0.999 0.001 What are your average winnings? The ordinary average of the two possible outcomes $0 and $500 is $250, but that makes no sense as the average winnings because $500 is much less likely than $0. In the long run, you win $500 once in every 1000 bets and $0 on the remaining 999 of 1000 bets. (Of course, if you play the game regularly, buying one ticket each time you play, after you have bought exactly 1000 DC-3 tickets, there is no guarantee that you will win exactly once. Probabilities are only long-run proportions.) Your long-run average winnings from a ticket are 1 $500- +$0- 1000 999 =$0.50 1000 or 50 cents. You see that in the long run the state pays out one- half of the money bet and keeps the other half.
Transcribed Image Text:Example 1 The DC Lottery Here is a simple lottery wager: the "Straight" from the DC-3 game of the DC Lottery offered by Washington, DC. You pay $1.00 and choose a three-digit number. The lottery chooses a three-digit winning number at random and pays you $500 if your number is chosen. Because there are 1000 three-digit numbers, you have probability 1-in-1000 of winning. Here is the probability model for your winnings: Outcome: $0 $500 Probability: 0.999 0.001 What are your average winnings? The ordinary average of the two possible outcomes $0 and $500 is $250, but that makes no sense as the average winnings because $500 is much less likely than $0. In the long run, you win $500 once in every 1000 bets and $0 on the remaining 999 of 1000 bets. (Of course, if you play the game regularly, buying one ticket each time you play, after you have bought exactly 1000 DC-3 tickets, there is no guarantee that you will win exactly once. Probabilities are only long-run proportions.) Your long-run average winnings from a ticket are 1 $500- +$0- 1000 999 =$0.50 1000 or 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 by multiplying 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,pk Here is a general definition of the kind of "average outcome" we used to evaluate the bets in Example 1. An expected value is an average of the possible outcomes, but it is not an ordinary average in which all outcomes get the same weight. Instead, each outcome is weighted by its probability so that outcomes that occur more often get higher weights.
Transcribed Image Text:The expected value of a random phenomenon that has numerical outcomes is found by multiplying 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,pk Here is a general definition of the kind of "average outcome" we used to evaluate the bets in Example 1. An expected value is an average of the possible outcomes, but it is not an ordinary average in which all outcomes get the same weight. Instead, each outcome is weighted by its probability so that outcomes that occur more often get higher weights.
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