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Mathematical Expectation

Mathematical Expectation. A friend offers you the chance to play the following game: You bet $2 and roll a die. If you roll a 6 you win $5 plus your bet If you roll a 5 you win your bet Any other number than you roll, you lose Should you play?. M.E. is short for mathematical expectation

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Mathematical Expectation

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  1. Mathematical Expectation

  2. A friend offers you the chance to play the following game:You bet $2 and roll a die. If you roll a 6 you win $5 plus your betIf you roll a 5 you win your betAny other number than you roll, you lose Should you play?

  3. M.E. is short for mathematical expectation • M.E. is the amount you will win or lose in a given situation or experiment. • M.E. = (probability of winning) x (net gain) + (probability of losing) x (loss) * Loss will be a negative number

  4. Is the game fair?Who is more likely to win? The game is fair when M.E. = 0 (the odds are 1:1 OR 50:50) The game is good for the player when M.E. is + The game is good for the dealer when M.E. is -

  5. We put all the information into a table A community organization holds a fundraising raffle and sells 6000 tickets at $5 each. • First prize is $10,000 • Second prize is $2000 • Third prize is $1000 • What is the expected value for this lottery?

  6. The Table M. E. = 1.67 + 0.33 + 0.17 - 4.99 M. E. = -2.82 The game favors the fundraiser, NOT the ticket holder

  7. How do you know if the game is fair? Mathematical Expectation = 0

  8. Example 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 This game favors Joe

  9. How do you know if the game is fair? Mathematical Expectation = 0 If the game favors the player it is NOT fair If the game favors the dealer it is NOT fair

  10. Example: Is this game fair? A game costs $4 to play. You roll a die. If you roll a 1, you keep your bet and win $12. If you roll a 2 or a 3 you keep your bet Anything else you lose your bet Yes it is fair M.E. = 2 + 0 – 2 = 0

  11. 6000 tickets at $5 each.First prize is $10,000 : Second prize is $2000: Third prize is $1000 M. E. = 1.67 + 0.33 + 0.17 - 4.99 M. E. = -2.82 How can we make this fair?

  12. 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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