State governments tend to ignore arguments about the possible bad effects of lotteries. That’s because, for the most part, they know enough about mathematical expectation to arrange that for each ticket purchased, the expected winnings—the total prize money divided by the number of tickets sold—is less than the cost of the ticket. This generally leaves a tidy difference that can be diverted to state coffers. In 1992, however, some investors in Melbourne, Australia, noticed that the Virginia Lottery violated this principle. The lottery involved picking 6 numbers from 1 to 44. Pascal’s triangle, should we find one that goes that far, would show that there are 7,059,052 ways of choosing 6 numbers from a group of 44. The lottery jackpot was $27 million, and with second, third, and fourth prizes included, the pot grew to $27,918,561. The clever investors reasoned, if they bought one ticket with each of the possible 7,059,052 number combinations, the value of those tickets would equal the value of the pot. That made each ticket worth about $27.9 million divided by 7,059,052, or about $3.95. For what price was the state of Virginia, in all its wisdom, selling the tickets? The usual $1.

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.

Excerpted from ‘The Drunkard’s Walk – How Randomness Rules our Lives’ by Leonard Mlodinow