By Rupert Li
One day during World War II, a group of U.S. generals approached mathematician Abraham Wald with data on where aircraft that returned from battle had been shot. They hoped Wald could devise a formula to reallocate armor to regions of the fuselage that were more frequently hit. However, Wald’s surprising insight disabused them of their implicit assumptions: critical modules such as the engine appeared fewer times in the data not because they were shot less often, but because the planes that were shot in the engine didn’t make it back home, so they ipso facto weren’t in the dataset. Mathematician Jordan Ellenberg of the University of Wisconsin-Madison, a two-time Putnam Fellow and two-time International Mathematics Olympiad gold medalist (both times with perfect scores), believes that this anecdote illustrates that the true essence of math is inquisitive thinking, rather than mere computation. In his video How Not to Be Wrong: The Power of Mathematical Thinking, adapted from his homonymous book, Ellenberg expounds upon how thinking mathematically can guide you to ask investigative questions toward deeper insights.
Ellenberg’s video focuses on one curious event: the Massachusetts state lottery from 2005 to 2011. It’s well-established that playing the lottery is a loss in the long run, so it might be puzzling to you, as it was to Ellenberg, why a group of MIT undergrads would pool together their money and hand-write hundreds of thousands of lottery tickets each day. Astoundingly, they continued for six whole years, exponentially growing their investment until they and two similar groups were purchasing 90% of all lottery tickets sold in Massachusetts. This finally ended after coverage from the Boston Globe shut the game down.
Something peculiar was going on. And in a manner accessible to all regardless of mathematical background, Ellenberg provides the mathematical reasoning behind why the lottery is typically not a good investment, but under the idiosyncrasies of this new lottery design, it was surprisingly feasible to reliably make money. Essentially, the lottery had a “roll-down” mechanism, where if the jackpot value became sufficiently high due to an insufficient quantity of prize-winning tickets, to incentivize a potentially disheartening lottery, the jackpot’s excess funds would be redistributed to valorize the smaller, easier payoffs.
Mathematical formulas have enabled us to decipher why this lottery was profitable. But the recurring motif throughout Ellenberg’s talk is that mathematics is not simply computing formulas, but also the art of probing under the surface and finding ensconced connections. A second look at this story prompts an additional question: how and why did the state permit these MIT students to exploit the system for so long? The answer, which Ellenberg found after subsequent investigation, turns out to be far more interesting than simply “the state messed up,” revealing unexpected insights into how the lottery and the government function.
So, the next time you encounter an intriguing occurrence, take a closer look. Your findings might just surprise you.