The Historical Context of Expected Value

The term ‘expected value’ is deeply rooted in the realms of gambling and betting. Before the advent of mathematical formalization, gamblers had an intuitive grasp of this concept. When faced with a bet on an uncertain outcome, they would assess whether it was a wise decision to proceed. Through experience and observation, gamblers developed an implicit understanding of expected value, allowing them to distinguish between advantageous and disadvantageous bets.

Consider how we often make quick decisions: Should I cycle in the rain or take a car? Fast food or a healthy salad? While the basic idea of expected value resonates intuitively, that’s where intuition can falter.

The formal mathematical concept of expected value began to take shape in the mid-1600s. This era, though seemingly distant, is relatively recent in the context of mathematical history. During this time, phenomena described by expected value often defied common sense.

The Concept of Annuities

Most people have some familiarity with insurance, and an annuity can be thought of as a reverse form of life insurance. An investor pays a lump sum for an annuity that disburses fixed amounts at regular intervals in the future—typically after retirement.

In 1692, England introduced the “Million Act,” which offered citizens annuities to help fund a war. This early version of annuities was priced uniformly, regardless of age. Consequently, a child would pay the same amount as an elderly individual with a much shorter life expectancy. Scientist Edmund Halley, after whom a comet is named, recognized this flaw and developed an age-adjusted pricing model using expected value.

To contemporary individuals, it seems obvious that a child would pay more than an older adult for an annuity. However, various iterations of such products existed previously, and it took a mind like Halley’s to identify the discrepancy in expected value. This highlights that the mathematical application of expected value is not always straightforward.

The Mathematical Basis of Expected Value

To grasp the mathematics behind expected value, let's examine a hypothetical betting scenario.

Suppose you are betting on a horse with a 10% chance of winning. The bookmaker offers you a bet at $10, with a potential payout of $200. If you lose, you receive nothing. What is the expected value of this bet?

To compute the expected value, we recognize two possible outcomes: win or lose. We multiply each outcome's value by its probability and sum the results:

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

This calculation suggests that the bet is valued at $20, yet the bookmaker is asking for $10. According to expected value, this bet seems highly favorable. However, it’s crucial to clarify some points before placing the wager.

Expected Value vs. Common Expectation

In our example, we identified only two outcomes: winning or losing. The calculated expected value of $20 doesn't align with any possible outcome. This raises an important point: the mathematical interpretation of expected value diverges from what we typically understand as “expected” in everyday language. Mathematically, expected value refers to what you can anticipate gaining on average after placing numerous bets on the same horse.

In simpler terms, the “expected” aspect relates to the statistical law of large numbers. If I were to explain expected value in layman's terms, I would describe it as the probability-weighted average of all possible outcomes.

Now that we've clarified this concept, let's discuss how to navigate the potential pitfalls when applying expected value.

Avoiding Common Pitfalls with Expected Value

As you may have noticed, many scenarios discussed are related to gambling or insurance, both of which adhere well to statistical predictions. However, many real-world situations do not fit this Gaussian framework, rendering expected value calculations potentially ineffective or even harmful.

Take the stock market, for instance. When expected value calculations are applied in this context, outcomes can become unpredictable. It’s concerning that “expected value” is often used in finance and other fields where it may not accurately predict outcomes. For those unfamiliar with the mathematical concept, this could lead to poor decision-making.

If you encounter “expected value” in any context, consider asking yourself:

“Does this expected value pertain to a phenomenon supported by the law of large numbers?”

If the answer is no, it may be best to disregard that figure. This simple rule can help you sidestep many traps associated with the concept of expected value.

Reference and credit: Jordon Ellenberg.

For further reading, you might enjoy: "Why Is The Hot Hand Fallacy Really A Fallacy?" and "How Imagination Helps You Get Good At Mental Math."

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