Probability
Probability studies uncertainty by assigning mathematical structure to random events, outcomes, likelihoods, and long-run patterns.
What probability studies
Probability is the mathematics of uncertainty. It gives a disciplined way to talk about events that may or may not happen, from coin tosses and weather forecasts to medical tests and machine learning predictions. The goal is not to remove uncertainty, but to measure and reason with it clearly.
Sample spaces and events
A sample space is the set of possible outcomes for an experiment or situation. Rolling a standard die has six outcomes; tomorrow's temperature has many possible values. An event is a subset of the sample space, such as rolling an even number or the temperature exceeding a threshold.
Rules of probability
Probabilities follow rules that keep reasoning consistent. The probability of an event is between 0 and 1. The probability of the whole sample space is 1. When events cannot happen together, the probability that either occurs is the sum of their probabilities. More general addition and multiplication rules handle overlap and dependence.
Conditional probability
Conditional probability updates a probability when new information is known. The chance of rain may change after a weather system is observed, and the chance that a test result indicates disease depends on how common the disease is. Conditional reasoning is powerful, but it is also where many everyday probability mistakes happen.
Random variables and distributions
A random variable turns outcomes into numbers, such as the total on two dice or the waiting time for a bus. Its probability distribution describes how likely each value or range of values is. Some distributions are discrete, with separate possible values, while others are continuous, spreading probability across intervals.
Expected value and variability
Expected value is a weighted average of possible outcomes, not necessarily the result that will happen next. Variance and standard deviation describe how spread out outcomes are around that average. Together, they help compare uncertain choices, such as investments, insurance risks, experiments, and forecasts.
Probability and statistics
Probability and statistics are closely related but not identical. Probability often starts with a model and asks what data or outcomes to expect. Statistics often starts with data and asks what model, pattern, or uncertainty might explain it. Modern inference moves back and forth between the two.
Why it matters
Probability matters because decisions often must be made before outcomes are known. It helps people interpret risk, avoid misleading intuition, design experiments, build forecasts, and understand uncertainty in scientific evidence. Used well, probability makes uncertainty less mysterious without pretending the future is guaranteed.