Expected Value (EV) is a mental model framework for making smarter decisions in uncertain situations. It evaluates choices based on potential outcomes, probabilities, and payoffs, ensuring decisions are rational, not emotional. In decision-making, EV acts like a compass. It balances risk and reward by multiplying an outcome’s probability by its impact – positive or negative. Billionaire investor Ray Dalio explains the goal:
“Normally a winning decision is one with a positive expected value, meaning that the reward times its probability of occurring is greater than the penalty times its probability of occurring, with the best decision being the one with the highest expected value.”
Thus, if a potential penalty is catastrophic, even 
high-reward opportunities may not be worth pursuing. By understanding EV, you can make statistically sound and repeatable decisions that will improve your long-term success. EV thinking also helps mitigate biases. Humans often overweight outcomes that feel vivid or frequent, even when they lack significance. For instance, people avoid low-probability, high-reward opportunities, overlooking their positive EV. Conversely, they persist with negative EV decisions, like continuing unprofitable projects due to sunk costs in time and resources.
Practically, EV isn’t about being right every time. It’s about following a process that delivers favorable results over many tries. Investors, poker players, and entrepreneurs use EV to identify opportunities where rewards outweigh the risks, even when success is slim. These thinkers embody what Nate Silver dubs The River.
His book, On the Edge, describes “The River” as a group of like-minded individuals – from poker players to crypto investors and venture-capital billionaires. This ecosystem shares a mindset centered on addressing decisions by using concepts like expected value, game theory, Nash Equilibria, and marginal utility. As Silver states:
“It’s a way of thinking and a mode of life. Most Riverians aren’t rich and powerful. But rich and powerful people are disproportionately likely to be Riverians compared to the rest of the population.” … ” I want you to understand that many powerful people and businesses think in expected-value terms – and they win more than they lose in the long run.”
What is Expected Value?
Expected Value (EV) is a mathematical tool used to evaluate outcomes under uncertainty. It calculates a decision’s average payoff by considering all possible outcomes, their probabilities, and impacts. This allows you to measure a decision’s long-term benefit or cost.
Formula:
EV = (Probability of Success x Reward) + (Probability of Failure x Penalty)
Breaking it down:
- Probability of Success: The likelihood of a favorable outcome.
- Reward: The payoff if the favorable outcome occurs.
- Probability of Failure: The likelihood of an unfavorable outcome.
- Penalty: The cost if the unfavorable outcome occurs.
Example: A bet has a 70% chance of losing $100 but a 30% chance of gaining $300.
- EV = (0.30 x $300) + (0.70 x -$100) = $90 – $70 = +$20 EV
- This positive EV suggests that, over time, taking this bet is statistically favorable.
Real-Life Applications of Expected Value
- Low-Probability, High-Reward Opportunities:
- If a $100 investment has a 20% chance of returning $1,000, its EV is +$120
- EV = (0.20 x $1,000) + (0.80 x – $100) = $200 – $80 = +$120 EV
- Avoiding Negative EV Bets: Parking Example:
- Parking cost = $5. Fine = $10. Probability of getting caught 33% (1/3).
- EV of not paying (2/3 x 0) + (1/3 x -10) = -$3.33 EV
- Example from Nate Silver’s book, On the Edge:
[T]he statistical model I built for FiveThirtyEight said there was a 71% chance that Hillary Clinton would win the presidency and a 29% chance that Donald Trump would win. Other statistical models put Trump’s chances at anywhere from 15% to less than 1%. And betting markets put them around 1 chance in 6 (17%). Trump won of course, by sweeping several Rust Belt swing states. The reaction of many people in the political world to this forecast was: “Nate Silver is a fucking idiot.” But from my standpoint – and from the standpoint of people in the River, the landscape of skilled gamblers and like-minded folks that I introduced in the prologue – this was a damned good forecast. It was a good forecast for a simple reason: if you’d bet on it, you would have made a lot of money. If a model says that Trump’s chance are 29% and the market price is 17%, the correct play is to bet on Trump – big. For every $100 you bet on Trump, you’d expect to make a profit of $74 or (0.71 x -$100) + (0.29 x $500) = +74.
4. Example from Annie Duke’s book, Thinking in Bets: Making Smarter Decisions When You Don’t Have All the Facts:
A few years ago, I consulted with a national nonprofit organization [that depended heavily on grant funding]. They provided me a list of all their outstanding grant applications and the award amounts applied for… I told them that I didn’t see how much each grant was worth in the information they provided. They pointed to the column of the award amounts sought. At that point, I realized we were working from different ideas about how to determine worth. This misunderstanding came from the disconnect between the expected value of each grant and the amount they would be awarded if they got the grant.
For example, if they applied for a $100,000 grant that they would win 25% of the time, that grant would have an expected value of $25,000 ($100,000 x .25). If they expected to get the grant a quarter of the time, then it wasn’t worth $100,000; it was worth a quarter of $100,000. A $200,000 application with a 10% chance of success would have an expected value of $20,000. A $50,000 grant with a 70% chance of success would be worth $35,000. Without thinking probabilistically in this way, determining a grant’s worth isn’t possible – it leads to the mistaken belief that the $200,000 grants is worth the most when, in fact, the $50,000 grant is.
Limitations of Expected Value
While EV is powerful, its use requires careful context. Nassim Taleb’s, The Black Swan, introduces the Ludic Fallacy – which highlights the dangers of applying game-like statistical models to the complexities of real-world situations. In controlled environments like casinos, probabilities are fixed and outcomes are calculable. In real life, however, the world is far less predictable. As Taleb writes, “The casino is the only human venture I know where the probabilities are known, Gaussian (i.e. bell-curve), and almost computable… In real life you do not know the odds; you need to discover them, and the sources of uncertainty are not defined.”
For instance, betting on the likelihood of rain might seem straightforward, but it involves a multitude of interdependent factors. Those factors include atmospheric conditions to unforeseen Black Swan events like sudden storms. Unlike Poker, where risks and rewards follow well-defined probabilities, many real-world scenarios fall under Knightian uncertainty (that which you cannot compute) – a state where risks cannot be quantified or reliably measured.
Applying EV without considering these uncertainties can lead to flawed decisions. As Taleb notes, attempts to “Platonify” uncertainty – forcing it into neat, mathematical frameworks – often creates a false sense of security. This misstep blinds decision-makers to crucial complexities and exposes them to risks they failed to anticipate. By acknowledging the limits of EV in unpredictable settings, individuals and organizations can make decisions that are not only statistically sound, but also contextually aware.
When to Use Expected Value in Decision-Making
EV is most effective when it’s used as part of a disciplined and iterative process. As Shane Parrish explains in a blog post titled, “Poker, Speeding Tickets, and Expected Value: Making Decisions in an Uncertain World, “Thinking in terms of expected value requires discipline and practice. And yet, the top performers in almost any field think in terms of probabilities.” Although EV might not feel natural at first, the benefits become apparent with practice, as EV improves the quality of both thinking and decision-making.
- Evaluating Low-Probability, High-Reward Opportunities:
- EV is useful when considering opportunities that carry significant upside potential despite low odds of success. For example, a trade or investment with a low probability of success might still carry a high expected value if the potential reward vastly outweighs the risk.
- Navigating Variability and Uncertainty:
- EV accounts for the range of possible outcomes, including wild deviations from the average. As Parrish notes, “Averages are useful, but they have limits, as the man who tried to cross the river discovered.” By incorporating deviations into calculations, EV highlights when a decision carries both manageable risks and significant potential benefits.
- Overcoming the Sunk Cost Fallacy:
- Many decisions are clouded by irrecoverable past investments of time, money, or resources. EV helps identify when to cut losses and move on. As Parrish explains, “Sunk costs push us toward situations with a negative expected value.” For example, a company might hesitate to cancel a doomed product launch despite clear evidence of failure, simply to justify prior investments. EV clarifies the better choice: avoid escalating commitment and redirect resources to more promising opportunities.
- Improving Long-Term Decision-Making:
- EV thinking fosters a process-oriented mindset. It’s not about being right in every instance; it’s about making decisions that, over many iterations, yield positive outcomes. This principle is particularly important in investing, where rational decision-making often involves choosing options with positive EV, even if they occasionally result in short-term losses