There’s [Probability] Theory, and Then There’s Reality

There’s [Probability] Theory, and Then There’s Reality

By Daniel Goldman | Trading Politics | 21 Oct 2019


Mathematicians sometimes forget that real life doesn’t always function like mathematical objects do.


Expected value is an important concept in mathematics, as well as topics of epidemiology and public health, economics and finance, project management, and much more. Expected value looks at how things tend to go, and is important in decision making. But sometimes those who use expected value ignore that real world events aren’t necessarily as forgiving as theoretical ones.

Expected Value

First, let’s cover the basic idea of expected value. For a random event, we never know exactly what outcome we’ll have, but we can still figure out a lot of information about random processes. The expected value is essentially the long term average of a random process, and it’s equal to the sum of the values a random variable can take, multiplied by the probability of it taking that value. The easiest way to understand the concept is through an example.

Consider a basic six sided die, with sides numbering from one through six. Assuming the die is fair, the probability of landing on any side is 1/6. And so the expected value is as follows.

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Another way to think about expected value is as a weighted average, where the weights are the probability of a random variable taking a specific value. Generally speaking, if two situations have the same expected value, we can think of them as in many way equivalent, in long run behavior. So for instance, if you received $3.50 per die roll, or you received the number of dollars indicated by the die roll, in the long run, it’s largely the same.

The Problem

A while back a student in one of the Coursera courses that I mentored questioned whether the two situations were really as similar as mathematicians might argue. Suppose that a disease afflicting 600 individuals has two cures, A and B. Cure A has a 1/3 chance of saving everyone, but a 2/3 chance of killing everyone, while cure B will certainly save 200 people and certainly kill 400. From a mathematical standpoint, both treatments have the same expected value. And yet, the two cures are very different, and the risk-reward analysis is actually very complicated.

Suppose that there are only 600 individuals in the entire population. Maybe it’s an endangered species. It’s true that there’s a 33% chance of saving every member of the population, and that seems like a good deal. But there’s also a 66% chance of killing the entire population, resulting in extinction. Meanwhile, the second treatment is guaranteed not to result in extinction.

Taking the Worse Deal

Some might argue that you’ll still want to select the choice with the greatest expected value and to never agree to a deal where the expected value is negative, as there would be a long run loss. For the most part, this idea holds. But the issue is that “long run behavior” does not apply in every case. Long run behavior requires being able to run as many trials as you want. So for instance, if there’s an extinction event, that’s it. For the same reason, people still get insurance, even though the expected value is negative.

Actuaries calculate risk of various payouts and your insurance premium is going to be higher than the expected value. However, without insurance, it may be impossible to pay, or it may bankrupt you, and so even though a mathematician might say that one should never take a deal with a negative expected value, odds are they’d still get insurance, or at least should, unless they’re really certain that they can save up enough money.

Getting it Right

The expected value is a powerful tool in risk-reward analysis, and it should not be ignored. For business operations with spare cash, it’s almost always best to go with the project with the highest expected value, and generally not a good idea to go with a project where the expected return, or expected earnings less cost, is negative. After all, if year after year we’re using available funds to generate more revenue, we’ll want the projects that will yield the best results, on average, and that’s generally determined by expected value.

However, in real life scenarios, we aren’t always guaranteed that we can keep playing the game over and over again, and so expected value has its limits. Taking on a risky project, or engaging in some activity that could be rewarding, but also potentially catastrophic, may not be the best option. Taking into account what each potential outcome really implies can be incredibly important, because we don’t always get a second chance.

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Daniel Goldman
Daniel Goldman

I’m a polymath and a rōnin scholar. That is to say that I enjoy studying many different topics. Find more at http://danielgoldman.us


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