A Hundred Dollars?
As James looked at Mr Lee, the class fell into a suspenseful silence. This was the first time Mr Lee had offered any sort of material reward, and everyone was curious about what it could be.
Smiling, Mr Lee addressed the class. "Now that you've learnt how to calculate probability, the next step is to understand the concepts of expected value and variance! Today, thanks to our intrepid volunteer, James, we will be able to get our first glimpse at these concepts."
Groans surrounded the classroom. Of course it was too good to be true. What else could be expected from a high school math teacher.
"First, James, here is your prize." Mr Lee said as he reached into his pocket and took out a shiny new black leather wallet.
At this sight, the class leaned forward in anticipation once again. A monetary prize was definitely something to be excited about.
James looked intently at Mr Lee's wallet. It was thin, sleek, and filled with possibilities.
Mr Lee opened his wallet and pulled out a thin, worn, one dollar bill.
"Here you go James. Well deserved." Mr Lee congratulated James once more.
Cheapskate! James thought. A perfect score in record time, and all I get is a measly dollar?
[ System agrees. ]
Despite his inner monologue, James reached out and took the dollar bill. A dollar was a dollar after all. This would put him back at a round $500 and make up for the dollar that the system had cheated him out of.
[ System does not cheat. System used skill and misdirection. ]
[ Oh, also... Current Money: $500. Current Winnings: $1. ]
As he accepted the prize money, the system updated his statistics. It looked like any form of winnings, including from a competition, would be recognized by the system. Either that, or the system was still hungry.
[ Hehe. ]
As James was about to turn around and head back to his seat, Mr Lee stopped him with an upraised hand.
"Not yet James. That was only the first part of the reward." Mr Lee said.
"Now, for the fun part. Expected value and variance. As part of this exercise, I guess we'll also get to see your risk appetite first hand. Now that you have the dollar, I'd like to offer you your choice of three different scenarios that you can use that dollar for." Mr Lee elaborated.
[ Expected value - what the actual value of a wager is. It is calculated as the sum of the probability of each possible outcome multiplied by the value of the outcome. ]
[ Variance - a measure of how different the possible outcomes are from what the actual value of the wager is. The higher the number, the more likely it is to be feast or famine. It is calculated as the sum of the probability of each possible outcome multiplied by the square of the value of each outcome before subtracting the square of the expected value. Often, in real world usage, the variance is then square rooted to give the standard deviation, which is a far more common metric and is more useful in calculating likelihoods given a large number of cases. ]
"The first scenario: I will flip a coin once. If it lands heads, I will give you one additional dollar. If it lands tails, you will lose the dollar in your hand."
"The second scenario: I will pick a card from a standard deck," said Mr Lee, making a deck of cards appear from his pants pocket, "If the card is a heart, you will win three additional dollars. If it is any other suit, you will lose the dollar in your hand."
"Finally, the jackpot scenario: I will write down a number between 1 and 100. If you can guess the number correctly, I will give you 99 additional dollars. If you guess wrong, you will lose the dollar in your hand."
"Which scenario would you like to choose?"
As James listened to the scenarios, his system provided a quick summary of each gamble.
[ Scenario One: Expected Value 0, Variance 1. ]
[ A fair fifty-fifty gamble. Double or nothing. ]
[ Scenario Two: Expected Value 0, Variance 3. ]
[ A chance to win more money, but you will lose three times as often as you win. ]
[ Scenario Three: Expected Value 0, Variance 99. ]
[ A small chance to get a large return, but expect to end up with nothing most of the time. ]
"All of these are roughly equivalent. If you could repeat them an infinite number of times, you would end up with exactly what you have right now. That is the expected value. Each of these have an expected value of zero." Sensing James' hesitation and the confusion of the class, Mr Lee quickly explained.
"Since you can only do these once, the difference arises out of the variance. Simply put, the variance is a measure of how different your outcomes could be. The larger the variance, the more different your outcomes are." Mr Lee went on to explain quickly.
James still hesitated. A dollar was not much in the grand scheme of things, but it would allow him to unlock the next step of the system. That was worth much more to him than a chance to double, triple, or even nonuple his money.
"Can I keep the dollar and decline?" James asked meekly.
At his words, mocking laughter erupted from the class.
"Jamie James, are you so poor that you'd rather have a dollar than a chance of ten?" Phil mocked. "Go for the big jackpot, at least then you can lose it all to me this weekend."
No one knew what the value of the single dollar was to James. It wasn't just $1, it was an opportunity to figure out more about the system his father had left him.
Mr Lee was stunned. He'd never seen anyone value a single dollar that highly before. He had expected James to try to go for the big one. A hundred dollars was a large amount to a high school kid after all. Even to him, if he lost, that hundred dollars would hurt. It was a decent chunk of his monthly salary. He had only offered that scenario because it was very unlikely that he would lose.
"Here James. Why don't I change the terms a bit. You can keep the dollar even if you lose." Mr Lee took pity on James and offered. He wanted to use the moment as a teaching opportunity for the class.
[ Scenario One: Expected Value 0.5, Variance 0.25. ]
[ Scenario Two: Expected Value 0.75, Variance 1.6875. ]
[ Scenario Three: Expected Value 0.99, Variance 97.0299 ]
[ System recommends, go for the big one! ]
"Ok sir. In that case, I'll go for the ninety-nine. Thank you." James politely responded. He could sense his system drooling at the prospects, and maybe with a large win, he would be able to unlock even more of the system.
Mr Lee smiled beatifically at James. "Good choice James. With the updated scenarios, that's the one that has the highest return."
Mr Lee then turned to the class before pulling out a quarter. "For demonstration's sake, I will show all three possibilities and their outcomes, and we can discuss later."
Mr Lee flipped the quarter in the air. The quarter tumbled through the air, landed on his desk, and bounced a few times.
Finally, the coin came to a rest, with the head of George Washington showing proudly on top.
"Heads it is. If you had chosen the first option, you would have doubled your money."
Mr Lee then picked up the deck of cards, gave them a quick shuffle, and flipped the top card over.
"The queen of hearts. If you had chosen the second option, you would now be three dollars richer!"
Finally, he turned around and scribbled on a small piece of paper before folding it and putting it in his left pants pocket.
"Now James, for the chance to exchange what is now your old, worn, dollar for my nice new spanking hundred dollar bill, what is your guess?" Mr Lee made his best imitation of a game show host. It wasn't very convincing.
James scrunched his face in deep thought. Given Mr Lee's personality, it was likely that he would have chosen something related to the gamble. Since the variance of the gamble was 97, he decided to go with that guess. It was better than taking a complete shot in the dark after all.
"97." James announced.
"Interesting choice," Mr Lee said, "Do you mind explaining why you chose that number?"
"It is the variance of the new scenario Mr Lee. I thought that you would choose something educational instead of a random number." James responded.
What scary insight! The smile on Mr Lee's face faded a little.