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Probability is a numerical measure of how likely an event is to occur, ranging from 0 to 1, or 0% to 100%. It is defined over events in a sample space and follows rules such as complement, addition, and multiplication. For example, the probability of heads when tossing a fair coin is 1/2, while the probability of at least one head in two tosses is 3/4. Probability may be interpreted objectively, as a long-run frequency or propensity, or subjectively, as a degree of belief based on available knowledge. Likelihood describes how well an observed result is supported by a particular hypothesis or parameter value. In Bayesian reasoning, prior probability is combined with a likelihood function to produce a posterior probability: posterior is proportional to prior times likelihood. Probability predicts outcomes given a model, whereas likelihood evaluates models or hypotheses given observed data. Probability theory is formalized through rules such as conditional probability, independence, mutual exclusivity, and Bayes' rule, and is widely applied in statistics, science, finance, artificial intelligence, risk assessment, and reliability analysis.
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