equalprobability

Equal probability, or equiprobable, describes a situation in which every elementary outcome in a given sample space is assigned the same likelihood. If the sample space has n mutually exclusive outcomes, each outcome has probability 1/n, and the probability of any event A is P(A) = |A|/n, where |A| is the number of outcomes in A.

In discrete contexts, equiprobability leads to the discrete uniform distribution. Examples include a fair six-sided die (each face has probability 1/6), a fair deck of 52 cards (each card has probability 1/52 when drawing one card at random), and random selections from a finite set where all items are treated identically.

In continuous contexts, equal probability is described by a uniform distribution on an interval [a, b]. In this case, the probability density function is constant: f(x) = 1/(b-a) for x in [a, b], and the probability of any subinterval [c, d] is (d-c)/(b-a). It is not meaningful to assign a nonzero probability to a single point in a continuum, since a single point has probability zero under continuous models.

Applications of equal probability include creating fair random models, teaching core concepts of probability, and serving as a baseline in statistical analyses. Real-world processes may approximate equiprobability but are often imperfect due to biases or measurement error. Empirical tests compare observed frequencies to expected frequencies under the equiprobable assumption.