Some People Have All the Luck

It is unusual to win a lottery prize worth $600 or more. No one we know has. But ten people have each won more than 80 such prizes in the Florida Lottery. This seems fishy. Someone might get lucky and win the Mega Millions jackpot (a 1-in-259 million chance) having bought just one ticket. But it’s implausible that a gambler would win many unlikely prizes without having bet very many times. How many? We pose an optimization problem whose answer gives a lower bound on any sensible estimate of an alleged gambler’s spending: over all possible combinations of Florida Lottery bets, what is the minimum amount spent so that, if every Florida resident spent that much, the chance that any of them would win so many times is still less than one in a million? If that amount is implausibly large compared to that gambler’s means, we have statistical evidence that she is up to something. Solving this optimization problem in practice hinges on two math facts: