would you play this game? roll a dice, and win $1000 dollars if you roll a 6

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

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

WOULD YOU PLAY THIS GAME?

• Roll a dice, and win $1000 dollars if you roll a 6

Page 2: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

PROBLEM: WHICH OUTCOME WOULD YOU BET ON?

Page 3: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

PROBLEM: WHICH OUTCOME WOULD YOU BET ON?

• Payout: Win $600

• Payout: Win $150

Page 4: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

ROLL A DICE 10 TIMES AND SEE WHICH EVENT WOULD PAY MORE

ON AVERAGE

• Payout: Win $600 • Payout: Win $150Number Rolled

Tally

1

2

3

4

5

6

Amount won:

Average:

Amount won:

Average:

Page 5: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

FINDING THE THEORETICAL EXPECTATION

• Payout: Win $600Probability = 1/6Value = $600Expectation = 1/6 x

600

=$100

• Try: Payout: Win $150Expectation = 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.

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

INVESTIGATING INVESTIGATING EXPECTATIONSEXPECTATIONS

Page 7: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

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.

Page 8: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

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.

Page 9: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

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?

Page 10: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

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?

Page 11: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

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?

Page 12: WOULD YOU PLAY THIS GAME? Roll a dice, and win $1000 dollars if you roll a 6

SUMMARYSUMMARY

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

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