Drunkard's Walk - Leonard Mlodinow [41]
The Australian investors quickly found 2,500 small investors in Australia, New Zealand, Europe, and the United States willing to put up an average of $3,000 each. If the scheme worked, the yield on that investment would be about $10,800. There were some risks in their plan. For one, since they weren’t the only ones buying tickets, it was possible that another player or even more than one other player would also choose the winning ticket, meaning they would have to split the pot. In the 170 times the lottery had been held, there was no winner 120 times, a single winner only 40 times, and two winners just 10 times. If those frequencies reflected accurately their odds, then the data suggested there was a 120 in 170 chance they would get the pot all to themselves, a 40 in 170 chance they would end up with half the pot, and a 10 in 170 chance they would win just a third of it. Recalculating their expected winnings employing Pascal’s principle of mathematical expectation, they found them to be (120/170 × $27.9 million) + (40/170 × $13.95 million) + (10/170 × $6.975 million) = $23.4 million. That is $3.31 per ticket, a great return on a $1 expenditure even after expenses.
But there was another danger: the logistic nightmare of completing the purchase of all the tickets by the lottery deadline. That could lead to the expenditure of a significant portion of their funds with no significant prize to show for it.
The members of the investment group made careful preparations. They filled out 1.4 million slips by hand, as required by the rules, each slip good for five games. They placed groups of buyers at 125 retail outlets and obtained cooperation from grocery stores, which profited from each ticket they sold. The scheme got going just seventy-two hours before the deadline. Grocery-store employees worked in shifts to sell as many tickets as possible. One store sold 75,000 in the last forty-eight hours. A chain store accepted bank checks for 2.4 million tickets, assigned the work of printing the tickets among its stores, and hired couriers to gather them. Still, in the end, the group ran out of time: they had purchased just 5 million of the 7,059,052 tickets.
Several days passed after the winning ticket was announced, and no one came forward to present it. The consortium had won, but it took its members that long to find the winning ticket. Then, when state lottery officials discovered what the consortium had done, they balked at paying. A month of legal wrangling ensued before the officials concluded they had no valid reason to deny the group. Finally, they paid out the prize.
To the study of randomness, Pascal contributed both his ideas about counting and the concept of mathematical expectation. Who knows what else he might have discovered, despite his renouncing mathematics, if his health had held up. But it did not. In July 1662, Pascal became seriously ill. His physicians prescribed the usual remedies: they bled him and administered violent purges, enemas, and emetics. He improved for a while, and then the illness returned, along with severe headaches, dizziness, and convulsions. Pascal vowed that if he survived, he would devote his life to helping the poor and asked to be moved to a hospital for the incurable, in order that, if he died, he would be in their company. He did die, a few days later, in August 1662. He was thirty-nine. An autopsy found the cause of death to be a brain hemorrhage, but it also revealed lesions in his liver, stomach, and intestines that accounted for the illnesses that had plagued him throughout his life.
CHAPTER 5
The Dueling Laws of Large and Small Numbers
IN THEIR WORK, Cardano, Galileo, and Pascal assumed that the probabilities relevant to the problems they tackled were known. Galileo, for example, assumed that a die has an equal chance of landing on any of its six faces. But how solid is such “knowledge”? The grand duke’s dice were probably designed not to favor any face, but that doesn’t mean fairness was actually achieved. Galileo could have tested his assumption by observing a number