Why there is always a winner, but it’s probably not you

I’d like to move away from the topic of lotteries, but not yet, because this is the window through which most people normally experience a very counter-intuitive mathematical law concerning probability and randomness. Indeed, the worldwide lottery business is primarily based on the assumption that the operators know this law and you don’t.

Setting up a truly-random and fair lottery is apparently harder than it would seem. Last August, Eddie Tipton, the security director of the Iowa-based Multi-State Lottery Association, was sentenced to 25 years in prison for rigging several state lottery results and trying to launder over \$2 million in winnings through friends and relatives. [1] At the end of the year, a coding problem in the South Carolina lottery caused a potential extra payout of \$19.6 million when it generated too many winning tickets. [2] And even more recently, the state of Connecticut messed up their New Year’s Super Draw and now need to run a second pass. [3]

I see numerous examples of “lottery mathematics” in nature, and I often use this oversimplifying term refer to a very common and natural form of probabilistic randomness where two conditions are met:

1. The odds of a “win” are very, very low (usually a very small fraction of one percent).
2. The number of “entries” qualifying for a “win” is very, very high (in the millions or billions within a short period of time).

These two conditions create results that are often counter-intuitive to the human brain, and thus hard for many of us to rationally accept. The mathematical Law of Large Numbers tells us that the statistical odds of any one person winning a well-constructed lottery are very tiny, but the odds of someone winning are nearly 100%. Quite the contrast. The law also tells us that, even though you have had a long streak of lottery losses, you still probably won’t win, and your past losses have zero effect on the future.

Especially when large jackpots loom, human emotion and rationality are at war here. We sniff at a “mere” \$1 million jackpot, yet stand in line for the \$500 million jackpot lottery, even though both would make us wealthy, and the odds for the latter win are, by design, likely 500 times worse than the former (which are already very, very bad). And more, the second contest is likely carefully designed mathematically to stretch an extra week or two in order to increase the anticipation and to goose sales.

The French mathematician Siméon Denis Poisson (1781–1840, pronounced “pwa-sawn”) changed the world with his description of an important time-based “Poisson probability distribution” and naming the “Law of Large Numbers” in 1837, although most of the world did not perceive the significance of his contributions at the time. [4]

Because of the Law of Large Numbers, every lottery director can bank quite reliably on how much the payout of the lottery is going to be over time (barring the screw-ups noted earlier). When the number of entries in a well-constructed lottery gets big enough, the total number of winners very is easy to calculate with this law within a very narrow range of probability. There will be a winner eventually, even if the jackpot has to roll over a couple of weeks. [5]

My favorite example of psychologically exploiting the Law of Large Numbers occurred in Michigan in the early years of their state-sponsored lottery in the mid-1980s. In the first round of the lottery, the payoffs were atypically frequent but very small. But if you lost this round, you were automatically entered into a second “Loser’s Lottery” with a much larger, but much lower-odds, payout. The television commercials showed people excitedly waving their losing tickets, shouting, “Oh boy, I lost again!” The commercial’s tagline was, “The more you lose, the more you win!” It was their most successful lottery to date.

In future posts, I will share some other examples of this law in action, ranging from the trivial, such as why you are amazed at the coincidence of seeing a particular vanity license plate, to the tragic reality that someone will die soon in a random shooting, but it will probably not be you. Oh, and the special case where you already are a “lottery winner.”

Notes:

1. Clayworth, Jason. “’I Certainly Regret’ Rigging Iowa Lottery, Says Cheat Who Gets 25 Years.” Des Moines Register, 22 Aug. 2017.
2. Fortin, Jacey. “Glitch in South Carolina Lottery Could Mean \$19.6 Million in Winnings.” The New York Times, 30 Dec. 2017.
3. “Human Error Prompts Second Super Draw Lottery Drawing.” NBC Connecticut, 1 Jan. 2018.
4. The Swiss mathematician Jacob Bernoulli (1655–1705) earlier proved a simpler version of this law. Poisson was able to generalize the law and gave it the name by which we now know it. I also have a post in the queue about “Poisson processes,” which explain much of the time-based randomness and probabilities that we encounter daily.
5. A very good, but technical explanation of this law is found at Khan, Salman. “Law of Large Numbers.” Khan Academy. Khan’s videos of math and science topics are always a good place to “nerd out” and learn something at the same time.

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