would you play this game? roll a dice, and win $1000 dollars if you roll a 6
TRANSCRIPT
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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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PROBLEM: WHICH OUTCOME WOULD YOU BET ON?
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PROBLEM: WHICH OUTCOME WOULD YOU BET ON?
• Payout: Win $600
• Payout: Win $150
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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:
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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.
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INVESTIGATING INVESTIGATING EXPECTATIONSEXPECTATIONS
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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.
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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.
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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?
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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?
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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?
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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