Fair Games/Expected Value

Definitions: The expected value of a game is the amount, on average, of money you win per game. The expected value (in terms of a game) is calculated as follows: E = ($ paid if you win) * (P(winning)) A game is a fair game when the cost of each game equals the expected value (what you put in, you get out).

GAME 1: You pay $3.00 to play. The dealer deals you one card. If it is a spade, you get $10. If it is anything else, you lose your money. Is this game fair? E = $10 * 13/52 = $10 * 1/4 = $2.50 $2.50 is the return, on average, of the game - - the expected value. $2.50 < $3.00, so it is not a fair game

GAME 2: A casino game costs $3.50 to play. You draw 1 card from a deck. If it is a heart, you win $10; If it is the Queen of hearts, you win $50. Is this a fair game? When there are multiple ways to win, the expected value is the sum of how much is won for each probability as follows. E = 10(12/52) + 50(1/52) = $2.31 + 0.96 E = $3.27 Since the disco earns 23 cents more, on average, than it pays out to you, it is not a fair game.

GAME 3: A player rolls a die and receives the number of dollars equal to the number on the die EXCEPT when the die shows a 6. If a 6 is rolled, the player loses $6. If the game is to be fair, what should be the cost to play? E = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5 (1/6) – 6(1/6) = 15/6 – 1 = 9/6 E = $1.50 (3/2 of a dollar) Charge $1.50 to play

GAME 4: Consider the above game with a modification. We would like to make a fair, FREE game. We will do this by charging a customer money if they roll a 1 as well as a 6. If all the rest is the same, what should we charge if they roll a 1? $0 = X(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5 (1/6) – 6(1/6) Solving for X we get = $8; We should charge $8 if they roll a 1.

GAME 5: This last game costs $1 to play. You are given a coin to flip. If you flip tails, the game ends. If you flip heads, you may flip again for a max of 5 flips. You will be paid $1 for each head. If all 5 flips result in heads, you win the $5 for 5 heads plus a $2 bonus. Is this a fair game?

Write out the cases: Cases Probability E E, calculated T ½ HT ¼ 1 * ¼ 8/32 HHT 1/8 2 * 1/8 HHHT 1/16 3 * 1/16 6/32 HHHHT 1/32 4 * 1/32 4/32 HHHHH (5+2) * 1/32 7/32 E = 33/32 = $1.03. This is NOT a fair game (but fun for YOU to play) the game will lose 3 cents per player, on average.