Chapter 6.3, Problem 50E

Illinois Lotto Exercises 49 and 50 refer to the Illinois Lottery Lotto game. (The data for these exercises were taken from Allan J. Gottlieb’s “Puzzle Corner” in Technology Review, February/March 1985.) In this game, the player chooses six different integers from 1 to 40. If the six match (in any order) the six different integers drawn by the lottery, the player wins the grand prize jackpot, which starts at $1 million and grows weekly until won. Multiple winners split the pot equally. For each $1 bet, the player must pick two (presumably different) sets of six integers.

In the game week ending June 18, 1983, a total of 2 million people bought $1 tickets, and 78 people matched all six winning integers and split the jackpot. If all numbers were selected randomly, the likelihood of having so many joint winners would be about ${10}^{-115}$. Can you think of any reason that such an unlikely event would have occurred? (Note: The winning numbers were 7, 13, 14, 21, 28, and 35.) What would be the best strategy in selecting the numbers to ensure that in the event that you won, you would probably not have to share the jackpot with too many people?