Statistics and the improbability generator

TL;DR – Events with low probability happen all the time, even ones with probability zero.

When I was in high school, a friend of mine vehemently told me: “Me with him? It’s more likely that donkeys start raining. It’s more likely that hell freezes over. It’s more likely that I enter a convent and never come out.” They have been married for fifteen years now.

Probability theory tells us that an event with low probability is less likely to happen than an event with high probability. For example, if you learned a bit of quantum mechanics (but not enough to know what you are talking about) you know that because of quantum tunneling you have a very low chance to go through a wall and appear on the other side. What you may not realize, though, is that you have a much greater chance of getting stuck in between. So, don’t try it at home.

We therefore treat events with low probability as things that almost never happen. But is it true that events with a low probability have a low chance of happening? Or do they happen all the time? Let’s find out in a probably unlikely thought experiment.

The improbability generator

You are a mad statistician. In the sense that you are really angry. Your funding has been cut because your proposal was deemed too risky. Your project unlikely to succeed.

Unlikely to succeed! You’ll show them something unlikely! You decided to create an improbability generator: a machine that creates very unlikely events all the time.

You work hard for a week and you have your prototype. A glorious machine that, when you push a button, prints out a digit from 0 to 9. Since each digit has the same probability of being chosen, the machine generates events with 10% probability 100% of the time.

The following week, you have a better idea: why not have it print two digits? Since each digit is chosen independently from the other, each combination of two digits has a chance of one in a hundred. The machine now generates events with 1% probability 100% of the time.

It takes you another week to realize that, actually, you know… you could print more digits? Maybe you can print a number with 3 digits (with a probability of 1 in a thousand) or maybe 9 digits (with a probability of 1 in a billion). But you settle to write a page with a 300-digit number. That means generating events with a probability of 1 over 10³⁰⁰, which is about the same probability of getting 996 heads straight when throwing a coin! You are sure that such improbability is likely to attract the attention of the funding agency.

In fact, the funding agency is so impressed that it funds further research to see whether you can increase the improbability factor of the machine by 10 times. What a challenge!

The chance of low probability events

While it is true that a particular event with very low probability is unlikely to happen, it is not true that events with very low probability, as a whole, are unlikely to happen. The chance of an event with very low probability to happen depends on how many of them there are. Each person has an equally very low probability of winning the lottery, but someone has to win.

One confusing aspect is that probability is something we assign. Each digit by itself had a 1-in-10 chance of being selected. But if we group them in a number of 300 digits, now each number has a 1-in-10³⁰⁰ chance of being selected. Nothing really changed in the process of selection. What grouping we consider is up to us, so we either have 300 relatively high probability events or one very low probability event depending on how we look at it.

This becomes even more confusing because events with probability zero can also happen. That is: events with probability zero are not, in general, impossible. Suppose we want to pick a number at random between zero and one. Since there are infinite numbers between zero and one, the probability of each number will be zero. But we are still going to pick one number: we are generating events with probability zero! What are the chances of that?

So, low probability events happen all the time and it’s a good thing. The chances that a person exactly like you would be born are extremely low. Maybe someday you will win the lottery. Or maybe a very strong gust of wind from Africa will lift an elephant, carry it over a great distance and drop it over your head. Wait, maybe that’s not a good thing after all…