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)
= $
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:
Cases
Probability E
E, calculated T HT
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 = $ This is NOT a fair game (but fun for
YOU to play) the game will lose 3 cents per player, on
average.