The secret to more accurate guessing with math

Suppose I showed you a box and asked you to guess what was inside without giving any further details. You might think this is completely impossible, but the nature of the container provides some information – for example, the contents must be smaller than the box, while a solid metal box can hold liquids and withstand temperatures that a cardboard box would have to deal with.

Is there a way to describe this limited information guessing process in a mathematically reasonable way? Clearly, there are things that cannot be reliably guessed—the toss of a coin, the roll of a die—and we call them random. But for everything else, a few useful tools can help you narrow down your guesses much better than picking an answer out of the ether.

A limited estimate is indeed an estimate, and they have a long history. Perhaps the most impressive early example is that of the ancient Greek philosopher Eratosthenes, who lived in Alexandria, Egypt in the 3rd century BC. Using a few simple ideas, he was able to estimate the circumference of the Earth with surprising accuracy. His exact method is lost, but we can reconstruct it thanks to texts written after his work.

Essentially, Eratosthenes knew that at noon on the summer solstice, the sun appeared to be directly overhead in the ancient city of Syene and cast no shadow on the well. Meanwhile, on the same day and time in Alexandria, the vertical bar cast a shadow at an angle of about 7 degrees, or roughly 1/50 of a circle. He knew that the distance between the two cities was 5,000 stadia, a unit of length, so it was estimated that the entire circumference of the earth must be 50 times that, or 250,000 stadia.

Eratosthenes made some approximations of geometry here, but we can ignore that. A little more complicated is that we don’t know the true value of the stadium. Eratosthenes is thought to have used something roughly equivalent to 160 meters. This gives us a circumference of 160*250,000 = 40,000 kilometers, which is remarkably close to the modern measurement of 40,075 kilometers. Of course, different values ​​for the stadium (ranging from 150 to 210 meters) will give you a different answer and a different level of precision depending on how generous we want to be to Eratosthenes.

The thing is, a few simple but reasonable calculations can give you a pretty strong estimate – measuring the planet without going around it. The master of the 20th century was physicist Enrico Fermi, who built the first ever nuclear reactor and played a key role in America’s Manhattan Project to develop the atomic bomb. He was present at the first detonation of such a weapon, the Trinity test, and tried to estimate the force of the explosion—no one was quite sure what it would be—by dropping small pieces of paper and watching how the explosion moved with them. like Eratosthenes, his exact technique has never been recordedbut his estimate that it was a 10-kiloton bomb is about half the actual value of 21 kilotons accepted for today’s Trinity yield. It’s not perfect, but at least it’s in the right ballpark.

Indeed, landing on the right pitch was kind of Fermi’s swing—he loved these kinds of back-of-the-envelope guesses so much that they’re now called Fermi problems. A classic example is a challenge that would set students: guess how many piano tuners there are in the city of Chicago. Starting with Chicago’s population (around 3 million), we could assume that the average household has four people, so there are 750,000 households. If one in five owns a piano, there are 150,000 pianos in Chicago. If we hire a piano tuner who can work on four pianos a week, they can get up to 1000 a year. So if the 150,000 pianos are serviced annually, there must be 150 piano tuners in Chicago.

The essence of this estimate is not that it is correct, but that it is limited by its incorrectness. We’ve made a number of assumptions along the way – but given that some will be overestimates while others will be underestimates, and assuming you’re not biased in one direction, then the errors are likely to be limited. If our calculations indicated that there were a million piano tuners in Chicago, for example, you could be pretty sure it was wrong.

While the Fermi estimate is a powerful technique for initial guesses, sometimes we gather new information that can help us refine our initial answer. Let’s go back to the box example I started with. If I took a blue ball numbered 32 out of the box, would that change your guess about its contents? You can assume that there are other balls inside the box, that some of them are blue and some have numbers – but is there a way to quantify this? Yes, thanks to Thomas Bayes, an 18th century statistician and church minister.

Bayes’ brilliant insight was to turn probability on its head, transforming it from a tool for understanding randomness—like the outcome of a coin toss—into a framework for measuring and reviewing uncertainty. He established an equation, Bayes’ theorem, for converting observations into evidence. It consists of four parts: prior, evidence, probability, and posterior. Let me explain each one in turn.

Prior is our basic premise. Let’s say I’m serving three flavors of ice cream at a party (chocolate, strawberry, and vanilla) and I want to know which one will be the most popular so I can stock up. A reasonable basic assumption is that flavor preferences are evenly distributed among people, with a third of the population liking each flavor. But then the party starts and I start to get nervous. The first 10 people went for chocolate – that’s my proof.

This is where it gets a little complicated. To define probability, I need to look at my original assumption. If taste preferences were truly the same, what are the chances of seeing 10 chocolates in a row? The answer is (1/3)^10, or about 1 in 60,000. That’s pretty unlikely, which suggests that my original assumption is probably wrong, and I need to update it to assume a much higher preference for chocolate, which in turn would give us a higher probability of seeing observed evidence. This update gives us back.

This sentence turns out to be extremely powerful. Back to my box example: the first ball I pulled out massively limits the possibilities of what’s inside. If I draw another ball, this one red and marked “50”, this narrows the possibilities even further – now you know that there are at least two colors of balls, and if you assume that they are numbered evenly, the total is likely to be small (under 100) rather than large (over a million). Each ball I draw gives you additional evidence that you can use to update your previous one each time.

One place you may have encountered Bayes’ Theorem without knowing it is in your email inbox. Spam filters used Bayesian reasoning, assuming that a certain percentage of emails are spam (prior), then using emails that you and your service provider flag as spam (evidence), combined with the possibility that certain words and phrases appear first in spam emails (probability), to determine which emails are actually spam (posterior).

Spam filtering shows why guessing isn’t a mathematical trick with boxes, but relevant to the real world. And using these techniques—Fermi estimation and Bayesian reasoning—is more important than ever in the world of pattern-matching AI like ChatGPT. As I wrote recently, the way modern AIs are built means they often seek to confirm rather than update or challenge your earlier ones, conforming to existing patterns without fully considering new evidence that doesn’t fit. Don’t let artificial intelligence guess incorrectly for you – learn to do it correctly yourself.

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