MostProbable

Most probable is a term used in probability and statistics to describe the outcome of a random variable that has the greatest likelihood. In statistical terminology this is the mode of the distribution: for a discrete random variable X with a probability mass function p(x), the most probable value is any x that maximizes p(x). For a continuous random variable with a probability density function f(x), the most probable value is the point(s) where f(x) attains its maximum. The mode can be unique or can consist of multiple values in a multimodal distribution or when ties occur.

The most probable value is not always the same as the mean or median. In symmetric unimodal distributions such as the normal distribution, the mode, mean, and median coincide. In skewed distributions they may differ; for example, a right-skewed distribution typically has mode less than the median and mean.

In statistical inference, the phrase is used in specific ways. The most probable parameter value given data in Bayesian analysis is the maximum a posteriori (MAP) estimate, which maximizes the posterior distribution. If the prior is uniform, the MAP estimate reduces to the maximum likelihood estimate (MLE). The concept also appears in practical forecasting as the most probable scenario or outcome.

Examples illustrate the idea: for a binomial distribution with parameters n and p, the most probable number of successes is floor((n+1)p); for a fair six-sided die, all six faces share the same probability, so there is no single most probable outcome.