1 math trick for better predictions

Warning, this is “I watched one YouTube video” level of expertise. Also, some graphs have truncated y-axis.

Predictions are fun. Will a dice roll four or greater? Will it rain tomorrow? Will this company be worth more money tomorrow, next month, next year? An event does or doesn’t happen. We get to predict an outcome.

If an NFL team wins six of their first seven games how many games will they win in total? Well 6/7 is ~85%, and there are seventeen games therefore they’ll win ~14.5 games. But in 2021 there was a team that won six of their first seven games and one math trick could predict it.

Pierre-Simon Laplace gives us the “rule of succession”. That sounds complicated but it’s simple: For any number of outcomes add one to the observed cases and two to the total cases.

Here are four coin flips: heads, heads, tails, heads. The observed rate for heads is 0.75 (3/4). The ‘Laplace’ rate for heads is 0.66 (4/6). Laplace’s addition shifts predictions away from ‘never’ and ‘always’. This is the secret. ‘Never’ and ‘always’ are rare for sequential events.

Here is what the Laplace rate looks like compared to the observed rate for eighteen coin flips.

Here is what the Laplace rate looks like compared to the observed rate for the “six of the first seven” football team, the 2021 Tampa Bay Buccaneers.

Laplace starts at .500. Tampa wins six of their first seven games (.857) but Laplace only increases to .777. Their final winning percentage was .764.

Then there’s the 2021 Detroit Lions, a team that lost their first eight games.

The Laplace rate doesn’t know anything. It doesn’t know coins are 50/50. It doesn’t know about Tom Brady. It doesn’t know the Lions are bad. It’s just a formula that slowly adjusts to extreme events.

Laplace (b. 1749- d. 1827) didn’t have the NFL, so he made predictions about something else, the sunrise. The observed rate is 1.00. The Laplace rate, after 10,000 observed sunrises, is 0.99990002. So you’re saying there’s a chance?

No. That’s a simple wrinkle. Laplace called the sunrise a special “phenomena” which “nothing at present moment can arrest the course of.”

Coin flips, dice rolls, and drawn playing cards are random and have an expected rate.

Sunrises are special phenomena and Laplace’s rate is less helpful.

Football outcomes are a mix. They’re like the sunrise, in that teams have inherent principles. They’re like coin flips in that predictions are difficult, a sign of randomness.

Math helps: relative vs absolute saving rates, people live longer the longer they live, what the mean age means, the vaccine friendship paradox, how many ants long is Central Park?, or how many rolls of toilet paper do the residents of Columbus Ohio use in a week?

Math can be simple. Technique (add one to the numerator, add two to the denominator) and a bit of explanation (extreme events are rare without explanatory phenomena) is all we need.

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