1. Mathematical Expectation Making the game fair

2. Make the Bet = X (we need to figure it out) Create the Table Under the column for outcome (O) subtract x from the gain Set up the Equation M.E. = p 1 o 1 + p 2 o 2 +p 3 o 3 etc Make the M.E. = 0

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

4. Let x = the bet ΩPOP X O 11/6 = 0.1710 -X(0.17) (10 – X) 61/6 = 0.175 -X(0.17) (5 – X) 2,3,4,54/6 = 0.67-X-0.67X 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 ΩPOP x 0 1rst place1/6000 = 0.00017$10000 - x 0.00017(10000 – x) 2 nd place1/6000 = 0.00017$2000 - x 0.00017(2000 – x) 3 rd place1/6000 = 0.00017$1000 - x 0.00017(1000 – x) 4 th place5997/6000 = 0.9995-x0.9995(-x) 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? 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