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# Fair Games/Expected Value

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))

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## Fair Games/Expected Value

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1. Fair Games/Expected Value

2. 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).

3. 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

4. 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.

5. 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

6. 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.

7. 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?

8. Write out the cases: 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.

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