1. 13.4
Expected Value

2. Essential Question:
How is expected value used to determine the fairness of a game of chance?

3. Expected Value:
The Expected Value is an average expectation per game if the game is played many times.
A wheel has 10 spots on it, 3 are red and 7 are blue. It cost $1.00 to spin the wheel and you are given $3.00 if it lands on red.
If we played the game 100 times how many times would we expect to win? Lose?
30 wins, 70 losses
How much would we win or lose each time?
Win $2.00, lose $1.00
For a total of +$60.00 and -$70.00 for winning/losing
So we expect to win -$10.00 over the long haul – correct?
How much do we “win” on average?
-$10.00/100 = -.10 per game

4. Definition of Expected Value:
A game gives payoffs a1 , a2 , … an with probabilitiesp1 , p2 , … pn. The expected value (or expectation) E of this game is:
E = a1p1 + a2p2 + … + anpn
A die is rolled, and you receive $1.00 for each point that shows. What is your expectation?

5. Example:
In Monte Carlo, the game of roulette is played on a wheel with slots numbered 0, 1, 2, … , 36. The wheel is spun and a ball dropped in the wheel is equally likely to end up in any one of the slots. To play the game, you bet $1.00 on any number other than zero. If the ball stops in your slot, you get $36. Find the expected value of this game.

6. Example 2:
You pay $1.00 to draw a card out of a deck. If you draw the ace of spades you win $100 what is your expected value?
An investor buys 1000 shares of a risky stock for $5 a share. She estimates that the probability the stock will rise in value to $20.00 a share is 10% and the probability that it will fall to $1.00 is 90%. Is this a wise investment?
The Athletic Booster Club offers a “reverse raffle” tickets are sold at $100 each, the winner gets $5000, what is the expectant?

7. Example 3:
A man “raffles” his $250,000 house. If the raffle tickets cost $ and he only sells 2000 tickets what is your expected value?
The probability of winning the lottery is 1 out of 13,983,816. What is the expected value of a $1.00 ticket if the winnings are 1,000,000?

8. 13.4 Classwork:
Pg : 2-18 evens

9. Revesiting Essential Question:
How is expected value used to determine the fairness of a game of chance?