workshop 7 solutions.pdf

workshop 7 solutions.pdf - Workshop VII Problems Matt...

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Workshop VII Problems Matt Hohertz November 2, 2016 Problem I “Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.” – E.T. Bell, “The Search for Truth.” Calculator allowed. In gambling, the house edge of a bet is defined as average winnings - bet amount bet amount where “average winnings” is the weighted average X all possible outcomes (amount of revenue for the outcome)(decimal probability of the outcome) Consider, for example, a game that pays 2 : 1 if I guess two consecutive coin flips correctly, and suppose that I can bet \$1 at a time on this game (the exact amount of the bet turns out to be irrelevant for calculating the house edge but we will make this assumption for simplicity). Then I have a 25% chance of receiving \$3 (\$2 plus my bet) and a 75% chance of receiving nothing; hence I will, on average, “win” ( . 25)(3) + ( . 75)(0) - 1 = - . 25 for every dollar I bet on this game. In other words, if I play this game often enough I will end up with a loss of a quarter for each dollar I wager (25% is a very large house edge even compared to state lotteries or the house edge on the “tightest” slot machines). When people say “the house always wins” in the context of casino gambling and lotteries, they are referring to the fact that nearly every possible bet results in an average net loss for for the player just like in the example – casinos would eventually go bankrupt were this not the case. 1

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Consider a casino game played as follows, with a jackpot that starts at \$0: 1. The player pays x dollars to the house in order to play. This amount is a fee and the house keeps it regardless of the outcome of the game.
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• Spring '11
• SOSA
• Calculus, Summation, 0.999...

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