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Fair Game: Mathematical Expectation in Betting Scenarios

Learn how to determine fair bet amounts using mathematical expectation. Calculate outcomes in games of chance like dice rolls to ensure fairness. Explore different scenarios by setting up equations and adjusting bets accordingly.

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Fair Game: Mathematical Expectation in Betting Scenarios

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  1. Mathematical ExpectationMaking the game fair

  2. Make the Bet = X (we need to figure it out)Create the TableUnder the column for outcome (O)subtract x from the gainSet up the EquationM.E. = p1o1 + p2o2 +p3o3 etcMake the M.E. = 0

  3. Determine the amount of the bet to make the following game fairRoll a dieIf you roll a 1 you win $10 If you roll a 6 you win $5Any thing else you lose your betn.b. the bet is included in the wins

  4. Let x = the bet M.E. = 1.7 – 0.17x + 0.85 – 0.17x – 0.67x 0 = 2.55 – x X = 2.55 The bet would have to be $2.55 for this game to be fair

  5. The Table M. E. = 1.7 - .00017x + .34 – 0.00017x + .17 – 0.00017x -.9995x M. E. = 0 0 = 2.21 – x X = 2.21 The ticket price would have to be $2.21 but it wouldn’t be much of a fundraiser would it?

  6. Changing the M.E. Joe bets $1 on the roll of a die If he rolls a 4: he wins $5 plus his bet 2: he wins $1 plus his bet 5: he gets his bet back Is it fair? M.E. = 0.85 + 0.17 + 0 – 0.5 = 0.52 Change the bet to make this game fair Change an outcome to make this game fair

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