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 bet If you roll a 5 you win your bet Any 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 ΩPOP x 0 1rst place1/6000 = 0.00017$99951.67 2 nd place1/6000 = 0.00017$19950.33 3 rd place1/6000 = 0.00017$9950.17 4 th place5997/6000 = 0.9995-$5-4.99 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? ΩPOP x O 41/6 = 0.1750.85 21/6= 0.1710.17 51/6= 0.1700 other3/6 = 0.5-0.5 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 ΩPOP x O 11/6 = 0.17122 2,32/6 = 0.3300 4,5,63/6 = 0.5-4-2 M.E. = 2 + 0 – 2 = 0 Yes it is fair

11. 6000 tickets at $5 each. First prize is $10,000 : Second prize is $2000: Third prize is $1000 ΩPOP x 0 1rst place1/6000 = 0.00017$99951.67 2 nd place1/6000 = 0.00017$19950.33 3 rd place1/6000 = 0.00017$9950.17 4 th place1/6000 = 0.00017-$5-4.99 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? ΩPOP x O 41/6 = 0.1750.85 21/6= 0.1710.17 51/6= 0.1700 other3/6 = 0.5-0.5 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