1. WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

2. PROBLEM: WHICH OUTCOME WOULD YOU BET ON?

3. Payout: Win $600 Payout: Win $150

4. ROLL A DICE 10 TIMES AND SEE WHICH EVENT WOULD PAY MORE ON AVERAGE Payout: Win $600 Payout: Win $150 Number RolledTally 1 2 3 4 5 6 Amount won: Average: Amount won: Average:

5. FINDING THE THEORETICAL EXPECTATION Payout: Win $600 Probability = 1/6 Value = $600 Expectation = 1/6 x 600 =$100 Try: Payout: Win $150 Expectation = 3/6 x 150 =$75 The expectation of an event is the product of the probability of an event, and the value of that outcome.

6. INVESTIGATING EXPECTATIONS

7. EXPERIMENTAL MEAN When you perform an experiment where each outcome has a certain value, we can calculate the experimental mean or average outcome for that experiment Eg) You roll a dice and win $1 for every even number rolled, and $2 for every odd number rolled. Play this game 10 times and find your experimental mean value.

8. THEORETICAL MEAN The theoretical mean value, or expected value, denoted E(X), is calculated by multiplying the probability of each event and its value, then finding the sum of these expectations. Eg) You roll a dice and win $1 for every even number rolled, and $2 for every odd number rolled. Find the expected value of this experiment.

9. FIND THE EXPECTED VALUE: 1.You flip a coin. If you flip heads you win $1. If you flip tails you lose $1. 2.Roll a dice. If you roll a 6 you win $100, otherwise you lose $1. 3.Pay $2 to draw two cards from a deck. You win $10 if you draw two face cards in a row. 4.Which of these games seemed fair? How can you tell?

10. LUCKY DOUBLE 7/11 Rules of the Game: Roll two dice at the same time. If you roll a sum 7, 11 or doubles you win $10 dollars. You pay $2 dollars to play the game. Your task: - Simulate playing this game 100 times and calculate your experimental mean value - Calculate your theoretical mean value - Explain : Who would you rather be, the player or the operator?

11. LOTTO 6/49 A ticket costs $5. You pick 6 different numbers between 1 and 49. On the day of the draw, 6 random numbers are chosen. If they all match the numbers on your ticket (in any order) then you win $10,000,000 What is the expected value of buying a Lotto 6/49 ticket?

12. SUMMARY What is expected value? How is it different from an experimental mean value? What does it mean for a game to be mathematically fair? When will the experimental mean value and the theoretical mean value be close to each other? p. 374 #3, 4, 8, 10, 12, 18 Challenge: 13