1 / 8

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))

marvela
Télécharger la présentation

Fair Games/Expected Value

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related