How To Really Understand Expected Value? - A cartoon that shows a stick figure with a fancy hat and a tie sitting inside a counter. A sign on top of the counter reads "$2 Window". Beside the counter a flap-sign reads "10 Horse Races Daily!" There are two other stick figures with money in their hands queued up at the counter. The first stick figure says "I want Horse-777 to win!"

Expected value is another one of those scientific terms like statistical significance that people misunderstand quite often. In this essay, I aim to clarify some of the common misconceptions about this concept.

We will begin by covering the historical origins of expected value. Then, we will look at mathematical formulation of expected value and cover how it differentiates itself from intuitive understanding. Finally, we will look at some common traps associated with the usage of expected value and explore how one could avoid these traps. Let us begin.

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The Origins of Expected Value

As with many other terms associated with uncertainty and probability, the term ‘expected value’ finds its roots in the gambling, gaming, and betting world. Before the days of mathematical formalization, gamblers had an intuitive understanding of expected value.

Given the option to bet on an uncertain outcome and a few other initial and boundary conditions, the typical gambler had to ‘estimate’ if taking the bet was a worthwhile venture. Through experience and empirical observation, gamblers had built an implicit model of expected value which could differentiate between favourable bets and unfavourable ones.

If you think about it, we intuitively do such calculations all the time to make split-second decisions: Is it worth riding the bike with a chance of rain or is it better to take the car or train? Fast food or salad? You get the picture. Although the fundamental concept makes sense intuitively, that is where intuition stops being helpful.

It was not until the mid-1600s that the mathematical notion of expected value started to develop. Although the 1600s sound like a long while back, in terms of mathematical history, this period is relatively recent. By this time, the phenomena that were described using the mathematical notion of expected value often turned out to be counterintuitive.

The Expected Value of Annuities

Most of us are aware of the concept of insurance. An annuity is “sort of” like life insurance in reverse. A prospective investor pays a lump sum today to purchase an annuity that would pay a fixed sum at regular time intervals (like monthly, yearly, etc.) in the future; after, say, the individual’s retirement.

In 1692, England came up with the “Million Act” which offered annuities to citizens to raise (drumroll please) a million pounds to go to war (ah, the good old times). This particular rudimentary “product” priced annuities at the same cost for any citizen regardless of their age.

This meant that a child who had many years of life left would pay the same for an annuity as would an elderly lady who probably had just a decade left. Scientist Edmund Halley (after whom a comet is famously named) figured out that this was incorrect and worked out an age-weighted version of annuity pricing. This calculation involved pricing using the mathematical notion of expected value.

This story features a key point to note: For modern folks like us, it sounds obvious that you charge a child more than an old lady for an annuity. However, the fact remains that many iterations of such products existed in the past. And it took a scientist of the caliber of Halley in the late 1600s to figure out that the expected value here was off. All this can just mean one thing: the mathematical usage of expected value is not obvious at all!

Since we are already on this topic, let us go a step further and see what the mathematical notion of expected value is about.

The Mathematics of the Expected Value

To understand the mathematics behind the notion of expected value, let us go back to its roots, and calculate the value of a hypothetical bet.

How To Really Understand Expected Value? — A cartoon that shows a stick figure with a fancy hat and a tie sitting inside a counter. A sign on top of the counter reads “$10 Window”. Beside the counter a flap-sign reads “20 Horse Races Daily!” There are two other stick figures with money in their hands queued up at the counter. The first stick figure says “I want Horse-222 to win!”
A Horse Racing Ticket Stand — illustrative art created by the author

Let us say that you are interested in betting on a horse in a horse race. Say that the horse of your choice has a 10% chance of victory (probability). The bookie offers you the bet for $10. If you win, you get $200. And if you lose, you get nothing ($0). The question is:

What is the expected value of this bet?

To calculate the expected value, the first thing we need to note is that there are only two possible outcomes for this bet: win or lose. To arrive at the expected value, we multiply the value of each possible outcome with its probability and sum them up. In this case, the expected value is as follows:

Expected Value = (10% * $200) + (90% * $0) = $20

So, the calculated expected value says that this bet is worth $20. But the bookie has offered it to you for $10. According to the expected value, this bet is overwhelmingly in your favour. So, you should consider placing the bet.

But before you jump in and place the bet, there are a few things that we need to clarify first.

Expected Value is not What You Expect it to be

In our hypothetical example, we had just two possible outcomes: win or lose. If you win, you would gain $200 and if you lose, you would gain $0. The expected value we calculated ($20) is not even one of the possible outcomes. So, just what does it mean then?

Herein lies a very important realization: the mathematical meaning of expected value has nothing to with what you would understand as “expected” value in common language. In mathematical terms, the term “expected value” refers to what you can expect to gain on average if you hypothetically placed a large number of bets on a large number of occasions on the same horse.

How To Really Understand Expected Value? — A cartoon showing an intense dialogue between two stick figures. The stick figure on the right says, “Expected value works in the context of law of large numbers!” The stick figure on the left agitatedly remarks, “Well, what am “I” supposed to “expect” then?!”
The conflict between mathematical language and common language — illustrative art created by the author

In other words, the “expected” part refers to the famous statistical law of large numbers. If you asked me to describe the mathematical notion of expected value in plain language, I would describe it as the probability-weighted arithmetic average value of all possible outcomes.

Now that we have clarified that, let us move on to potential traps when using the notion of expected value and how you might avoid them.

How to Avoid Potential Traps When Using Expected Value

If you had been perceptive, you would have noticed a common theme throughout this essay. All of the scenarios we have been dealing with are either gambling-/betting-related or insurance-related. These are all gaussian environments that yield favourably to statistical predictions.

Many real-world phenomena are not gaussian in nature. This means that computing the expected value for such phenomena is at best, useless (at worst, harmful). Consider the stock market for instance. When any sort of expected value calculation is done in the context of the stock market, things are likely to get dicey.

It is a shame that the term “expected value” is calculated and used in finance and other areas where it is ill-equipped to ‘expect’ or ‘predict’ anything. If the end user is uninitiated about the mathematical concept of expected value, it is a recipe for disaster. In such cases, the “professionals” engage in immoral behaviour, in my opinion.

If you are an end-user who encounters the term “expected value” in any context, ask yourself the following question:

“Does this expected value describe a phenomenon that is backed by the law of large numbers?”

If the answer is no, then I suggest that you best disregard this number! This simple thumb rule alone would help you avoid a whole host of traps surrounding the usage of expected value.

Reference and credit: Jordon Ellenberg.

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Further reading that might interest you: Why Is The Hot Hand Fallacy Really A Fallacy? and How Imagination Helps You Get Good At Mental Math?

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