What exactly is probability, and why does it matter? Probability is the mathematical measure of how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). Rather than just an abstract mathematical concept, probability gives us a precise language for talking about uncertainty—whether we’re predicting election outcomes, calculating casino odds, or deciding which medical treatment to pursue.
Understanding probability and its related concepts allows us to move beyond guesswork and make better decisions based on the actual likelihood of different outcomes. Below, we’ve put together ideas from On the Edge by Nate Silver and Principles: Life and Work by Ray Dalio to explore basic concepts in probability.
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What Is Probability?
Probability is a measure of the likelihood or chance that a specific event will occur. It’s expressed as a number between 0 and 1, where 0 indicates the event definitely won’t happen, and 1 indicates the event definitely will happen. Probabilities closer to 1 mean the event is more likely to occur, while probabilities closer to 0 mean the event is less likely.
In On the Edge, Nate Silver explains that probability is a powerful tool because it gives us a way to be precise about uncertainty. Probability allows us to understand and quantify uncertainties in various situations, from predicting election outcomes to assessing the risks associated with different scenarios. When people calculate the probability of a particular outcome happening, most of the time they’re trying to determine the chances of what they’d consider a favorable outcome against the chances of what they’d consider an unfavorable outcome.
(Shortform note: Why do we express probability as a number between 0 and 1? This scale is more intuitive than it might seem: It’s just another way of writing percentages, where 0 means 0% chance (impossible) and 1 means 100% chance (certain). We use decimals instead of percentages because they’re easier to work with mathematically. For instance, 0.5 means there’s a 50-50 chance, like flipping a coin. This standardized way of expressing probability helps us communicate about uncertainty in a consistent way. However, some mathematicians argue we should never assign probabilities of exactly 0 or 1 to real-world events, since nothing is truly impossible or absolutely certain.)
The ability to precisely quantify probability forms the foundation of some of the most powerful, yet basic, concepts in probability that Silver writes about: the concepts of expected value, Bayesian probability, and game theory.
1. Expected Value
A foundational idea in the study of probability is expected value. Expected value represents the average outcome of a random variable over a large number of trials. It’s calculated by multiplying each possible outcome by the probability of its occurring, and then summing up all these products. For example, you can use expected value to estimate the average monetary value you can expect to gain or lose in a casino game, based on the probabilities of different outcomes. In other words, expected value tells you the amount of money you can theoretically expect to win or lose over the long run.
Silver explains that expected value forms the foundation of our modern understanding of economics and the kind of economic analysis called behavioral economics. If every person is a rational thinker who is trying to maximize the expected value of our choices, gambles, and investments, then that gives a predictable order to how we’ll tend to behave under conditions of uncertainty. This means that our choices are predictable—which can be useful both for researchers looking to understand how the economy works and for investors who are seeking to maximize their returns by going in a direction that other people won’t.
| The Human Side of Expected Value While Silver’s explanation of expected value calculations offers a rigorous framework for decision-making, hedge fund founder Ray Dalio’s experience shows how this mathematical approach can be enriched by human wisdom. In Principles, Dalio advocates using expected value calculations (multiplying potential outcomes by their probabilities), but demonstrates how they often work best when combined with other forms of knowledge. When faced with his own cancer diagnosis, for instance, Dalio received six different medical opinions with vastly different probabilities attached to each outcome. Rather than simply averaging these probabilities, he used them as one input among many, weighing them against his personal principles and risk tolerance to make his decision. This illustrates how successful decision-makers often combine probabilistic thinking with other forms of intelligence: They use expected value calculations to clarify their options, while drawing on experience and judgment to navigate complex situations. As Dalio shows, the art of decision-making lies not in replacing human wisdom with pure calculation, but in finding ways to make them work together. |
2. Bayesian Probability
A cornerstone of the mathematical approach that Silver takes to statistics—in both poker and politics—is Bayesian probability. This approach involves updating your beliefs about the likelihood of an event occurring based on new evidence. It allows you to adjust your probability estimates as you receive more information, combining your prior knowledge with new data to arrive at better predictions.
For example, imagine you’re playing poker against someone new. You might start with a general assumption that they’re an average player (your “prior knowledge”). But after watching them play several hands skillfully, you update your probability estimate of them being an expert player. With each new piece of information—how they bet, which hands they fold, how they react to bluffs—you continue refining your estimate of their skill level.
This approach differs from traditional (or “frequentist”) probability, which would only look at how often certain outcomes occur in repeated trials. A frequentist approach might only consider the percentage of expert players in the general population, while Bayesian thinking allows you to incorporate specific observations about this particular player to make a more informed judgment.
(Shortform note: While Silver presents Bayesian probability primarily as a tool for human decision-making, research suggests the ability to update beliefs based on new evidence may be fundamental to how complex brains process uncertainty. Scientists studying cetaceans use Bayesian models to study everything from whale populations to killer whale hunting success. They’ve also found that cetaceans themselves appear to have some understanding of probability: Dolphins can signal their level of uncertainty about their knowledge, suggesting they grasp when they don’t have enough information to make confident predictions.)
3. Game Theory
Gathering new data to incorporate into your analysis often involves taking in new information about what other people are doing. Game theory is the mathematical study of the strategic behavior of two or more agents (players) in situations where their actions impact one another. It seeks to predict the outcome of interactions and model the optimal strategies for each player to maximize their expected value while considering the actions of other players. Nash equilibrium, named after mathematician John Nash, is a fundamental concept in game theory that describes a situation in which each player in a game has chosen a strategy that’s optimal given the strategies chosen by all other players.
In other words, in a Nash equilibrium, no player has an incentive to unilaterally change their strategy because doing so wouldn’t benefit them given the strategies that the other players have chosen. Understanding the Nash equilibrium helps us predict how rational decision-makers should behave when faced with situations where their choices impact and are impacted by the choices of others. By identifying the Nash equilibrium, game theory enables us to model and analyze strategic behavior, predict outcomes, and determine the most advantageous strategies for each player in a given situation.
Silver explains that researchers use game theory in economics, the social sciences, and computer science to predict how people will behave. Game theory is also crucial for gamblers: Being a good poker player involves predicting what your companions are going to do—while remaining unpredictable yourself. Randomization and deception enable players to gain an advantage by keeping their opponents guessing and preventing them from easily predicting their strategy.
| The Biology of Game Theory Silver’s presentation of game theory as a tool for understanding strategic behavior finds confirmation—and elaboration—in bacterial research. Scientists have discovered that some bacterial interactions follow game theory’s predictions with remarkable precision: Certain strains of E. coli compete in a perfect rock-paper-scissors pattern, where one strain produces a toxin, another is resistant to it but grows slowly, and a third grows quickly but is vulnerable to the toxin. Each can outcompete one of its rivals but is vulnerable to the other, creating a perpetual cycle that game theory predicts. Research on “selfish” bacteria (which continue growing when resources are scarce) and “cooperative” bacteria (which slow their growth to preserve resources) has revealed how game theory can help us understand more complex competitive dynamics. While Nash equilibrium calculations suggest selfish bacteria should dominate, experiments show that cooperative bacteria often survive by organizing themselves spatially—suggesting that game theory can be expanded to account for how physical and social structures affect competition. Game theory’s insights extend beyond human behavior to illuminate patterns throughout nature, while also revealing new dimensions of balancing competition and cooperation. |
These mathematical concepts—probability, expected value, Bayesian reasoning, and game theory—form a shared language that enables people to analyze risk and make decisions under uncertainty. For some people, this quantitative approach becomes more than just a toolkit: It becomes a way of seeing the world.
Learn More About the Basic Concepts in Probability
If you found this article interesting and you want to learn even more about basic concepts in probability, how it learns, and the different systems, you can read the full guides to the books mentioned above:
