Three investment bankers met for a power lunch.
"Guys," said the first banker, "I just had a billion-dollar idea."
"You'd better tell us," said the other two bankers simultaneously. They could smell the scent of their impending bonus checks, even over the delicious aroma ascending from the filet mignon.
"It's a simple matter of probabilities," said the first banker, adjusting his tie. "You know the lottery, right? Suppose I go out to the store and buy a ticket. What are the two possibilities?"
"Well," said the second banker, furrowing his brow, "I can think of one. There are really two?"
"I'm stumped too," said the third banker. "You'd better tell us."
"Well," said the first banker, feeling very proud of his insight, "either I win, or I lose."
"Oh, of course," said the second banker. "I had thought of both of those. I just counted them wrong."
"Anyway," said the first banker, "there are two possibilities. As everyone knows, that means the probability of each is 1/2! So if I play the lottery, the probability that I win is 1/2!" He pounded his fist on the table.
"Gentlemen," he said, "I think I'm going to open a new fund. I'll call it 'Modern Probabilistic Investment Fund.' I can offer you the opportunity to contribute some capital for a minority stake..."
"Wait a moment," said the second banker, interrupting the negotiations. "Suppose I were to tell you how you could make twice as much money. Could I be an equal partner in the fund if I did that?"
"I suppose so," said the first banker. "What's your idea?"
"Well," said the second banker, "we've just seen that if you buy one lottery ticket, the probability that you win is 1/2. So if you buy a second lottery ticket, the probability that you'll win with that one is also 1/2, right?"
"Of course," said the first banker. "Go on."
"Now, what is 1/2 plus 1/2?" asked the second banker.
"Seven!" said the third banker. "Wow! A probability of seven! Who ever heard of such a thing?"
"I don't think that's right," said the first banker, who had taken a calculator from his briefcase. "My calculator says that the answer is 1."
"Good," said the second banker. "That's what I got too. So if you buy two tickets, the probability of winning is 1. That's twice as much as if you had bought only one ticket!"
"Brilliant!" said the first banker. "I knew you guys wouldn't let me down. So, about that fund. I'll be the chairman, and you can be the president."
"No," said the second banker. "I want to be the chairman. You can be the president."
"No," said the first banker. "I'm definitely going to be the chairman."
After this had been going on for some time, the third banker at last broke in.
"Wait a minute," he said. "I just thought of a problem with your idea."
"Hey, enough of your negative thinking," said the first banker. "That's the kind of thing that brings down companies, you know."
"It couldn't hurt to hear it, though," said the second banker. "Let's see what he says before we dismiss it. It's good to keep an open mind."
"Thank you," said the third banker. "Now, when you buy two lottery tickets, would you buy two with the same numbers selected, or with different numbers?"
"Well, obviously with different numbers," said the first banker scornfully. "If they had the same numbers, then you'd split the pot with yourself if you win, so it would just be the same as buying one ticket. We're not as stupid as you think."
"That's not the problem," said the third banker. "But now consider the possibilities. Since the tickets have different numbers, only one of them can win, right?"
"Of course," said the second banker.
"OK," said the third banker. "So there are three possibilities. Either the first ticket wins, or the second ticket wins, or neither ticket does. That's three possibilities, and you win in two of them. So the probability of winning is actually 2/3!"
"So what?" said the second banker. "2/3 is still better than 1/2. And even with a probability of 1/2 we'll still make enough money."
"There's no such thing as enough money," said the first banker. "But whatever. I don't see the problem."
"Well," said the third banker, "Let's say the jackpot is one million dollars. If you leverage the one-ticket method, you get an average of 500,000 dollars, but you have to pay one dollar to do it. So your return on investment is 50,000,000%. If you leverage the two-ticket method, you get an average of 666,667 dollars, but you have to pay two dollars to do it, so your return on investment is only 33,333,333%. So the best method is the one-ticket method, and not the two-ticket method!"
"Whatever," said the second banker. "I still get to be chairman."
"No, I do," said the first banker.
"Wait a moment," said the second banker. "If the one-ticket method is better than the two-ticket method, then wouldn't the best method be the ZERO-ticket method? After all, one is less than two, and zero is even less than one! You could make even more money that way!"
"I don't know," said the third banker. "You just went way beyond my mathematical abilities. But I'll call my quant. He's got a PhD in math, so I'm sure he'll know the answer."
So the third banker got out his Blackberry and dialed a number. "Vitaly," he said into the phone, "we've been having a power lunch and we need to know the answer to a question."
There was something said on the other end of the phone that neither of the other bankers could hear.
"Here's the question," said the third banker. "Is it true that the best number of lottery tickets one can buy is zero?" There was a tense pause. Even the waiters froze in their tracks, waiting. The fate of the city, the planet, and even the entire known universe seemed to hang in the balance.
"Vitaly says yes," said the third banker. "The best number of lottery tickets to buy is zero."
"But," said the first banker, "if we don't buy any lottery tickets, then we won't make any money at all!"
"That's too bad," said the second banker.
"A shame," said the third banker.
They turned sadly back to their filet mignon.