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Category: Models
Type: Probability Model
Origin: Mathematics, 17th century-present
Also known as: EV, Mathematical Expectation, Mean Value
Quick Answer — Expected value is the weighted average of all possible outcomes of a decision, where each outcome is multiplied by its probability of occurring. It tells you the average result you can expect over the long run from any uncertain situation.

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What is Expected Value?

Expected value (EV) is a fundamental concept in probability theory and decision science that quantifies the average outcome of an uncertain event. By combining the value of each possible outcome with its probability, EV provides a single number that represents what you can expect to gain—or lose—on average over many repetitions.
“Expected value is not about predicting a single outcome, but about understanding the long-term average of repeated decisions.”
For example, if you have a 50% chance to win 100anda50100 and a 50% chance to win nothing, the expected value is (0.5 × 100) + (0.5 × 0)=0) = 50. This doesn’t mean you’ll win 50eachtime—instead,ifyourepeatedthisbetmanytimes,you′daverage50 each time—instead, if you repeated this bet many times, you'd average 50 per attempt.

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Expected Value in 3 Depths

  • Beginner: Multiply each possible outcome by its probability and add them up. A 10% chance to win 1,000hasanexpectedvalueof1,000 has an expected value of 100.
  • Practitioner: Use expected value to compare decisions with different risk profiles. A higher expected value doesn’t guarantee a better outcome in any single case, but it does over many repetitions.
  • Advanced: Consider utility—the subjective value of outcomes to you. A $100 gain might be worth more to someone with little money than to someone wealthy. EV calculations should factor in your utility function.

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Origin

The concept of expected value emerged from the study of games of chance in the 17th century. French mathematicians Blaise Pascal and Pierre de Fermat developed the foundational probability theory while solving gambling problems in the 1650s. The term “expected value” itself was coined later, but the mathematical principle became the cornerstone of probability theory. The concept gained further importance with the development of statistics and decision theory in the 20th century. Today, expected value is essential in fields ranging from finance and insurance to poker strategy and business investment analysis.

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Key Points

1

EV requires considering all outcomes

A complete EV calculation must include every possible outcome, not just the most likely ones. Missing an improbable but high-value outcome can dramatically change the calculation.
2

Probability and value must both be considered

A small chance of winning millions might have lower EV than a high probability of winning a modest amount. Both probability and payoff matter.
3

Long-term average, not single-event prediction

EV describes what happens on average over many repetitions. In any single instance, the actual outcome may differ significantly from the expected value.
4

Risk tolerance affects decision-making

Even with perfect EV calculations, people with different risk tolerances may make different choices. Understanding your risk profile is essential.

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Applications

Investment Analysis

Investors use expected value to evaluate opportunities. A startup investment might have 90% chance of failure (losing all money) but 10% chance of 20x returns, yielding positive EV.

Poker Strategy

Professional poker players calculate the expected value of each decision—whether to bet, call, or fold—based on the probability of winning and the pot size.

Insurance Decisions

Insurance premiums are calculated using expected value. The expected medical cost determines what you should be willing to pay for coverage.

Business Decisions

Product launches involve uncertain outcomes. EV analysis helps quantify the potential return of launching a new product versus the cost of failure.

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Case Study

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Jeff Bezos’s Decision Framework

Before founding Amazon, Jeff Bezos developed a systematic approach to evaluating career opportunities that deeply relied on expected value thinking. Working at hedge fund D.E. Shaw in the early 1990s, he faced a choice: stay in a lucrative Wall Street career or pursue the risky venture of selling books online. Bezos created a “regret minimization framework” that he described as picturing himself at age 80 and asking which decision would minimize regret. But underlying this was a clear-eyed EV calculation: the internet was growing at 2,300% annually, and the potential market for online bookselling, while uncertain, offered astronomical upside if successful. The expected value of staying at D.E. Shaw was highly positive but capped—good salary, steady career. The expected value of launching Amazon was highly uncertain but had no upper bound. Bezos calculated that even with high failure probability, the massive potential upside made the EV positive. This EV thinking—“maximizing the number of life dimensions in which you can win”—became central to Amazon’s corporate philosophy. The company has repeatedly entered businesses with high failure risk but massive potential upside: AWS, Kindle, Prime. The lesson: understanding expected value helps you take calculated risks that compound over time into enormous advantages.

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Boundaries and Failure Modes

Expected value analysis has important limitations:
  1. Rare events are hard to probability-weight: Black swan events—extremely unlikely but high-impact outcomes—are difficult to incorporate accurately into EV calculations.
  2. Utility is non-linear: A 1milliongaindoesn′thave10xthevalueofa1 million gain doesn't have 10x the value of a 100,000 gain for most people. EV assumes linear utility, which may not reflect real preferences.
  3. Probability estimates are often wrong: EV is only as good as your probability estimates. Overconfidence in estimating probabilities leads to poor decisions.
  4. Repeated trials required: EV only manifests over many repetitions. If you can only try once, a positive-EV bet with 90% chance of losing everything might still be too risky.

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Common Misconceptions

EV describes the average over many repetitions, not what will happen in any single instance. A positive-EV bet can still lose.
Positive EV often comes with significant risk. A 1% chance of winning 1millionhaspositiveEV(1 million has positive EV (10,000) but 99% chance of winning nothing.
Real-world decisions often involve unknown probabilities and unquantifiable outcomes. EV is a guide, not a precise formula.

Utility Theory

How people assign subjective value to different outcomes, beyond monetary calculations.

Risk Assessment

The process of identifying and evaluating potential risks in decisions.

Decision Tree

A visual model that maps decisions and their possible consequences.

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One-Line Takeaway

Make decisions based on expected value, not just likelihood or payoff alone. Over many repetitions, EV-driven choices compound into superior outcomes.