Probability
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Probability is a branch of mathematics that measures how likely a particular event is to occur. The probability of an event is expressed as a number between 0 and 1: 0 means the event cannot happen, 1 means it is certain to happen. It can also be expressed as a percentage—for example, 0.5 is equivalent to 50%.
Probability forms the foundation of statistics, insurance, quantum physics, machine learning, and business decision-making. When outcomes cannot be predicted with certainty, probability provides a mathematical way to assess risk and make forecasts.
Basic Concepts
An experiment is an action with different possible outcomes, such as flipping a coin. A sample space is the set of all possible outcomes—for example, {heads, tails}. An event is a subset of those outcomes. If all outcomes are equally likely, the probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Important Laws
The sum of the probabilities of all outcomes in a sample space equals 1. The probability of an event not occurring (the complement) is 1 minus the probability of the event. If two events cannot occur together, the probability of either one happening is the sum of their individual probabilities. If two events are independent—one does not affect the other—the probability that both occur is the product of their probabilities.
Conditional Probability and Bayes' Theorem
Conditional probability is the probability of event A given that event B has occurred. Bayes' theorem describes how a prior belief is updated to a posterior belief when new evidence is obtained. This approach is widely used in medical diagnosis, spam filtering, and artificial intelligence systems.
Interpretations of Probability
Two major interpretations exist. The frequentist interpretation views probability as the long-run proportion of times an event occurs when an experiment is repeated many times. The Bayesian interpretation views probability as a measure of a person's confidence in a claim, which is updated as new evidence emerges. Both use the same mathematical framework, but assign different meanings to the numbers.