Every random experiment carries a map of likeliness
A fair coin is not just heads or tails—it is a distribution that splits probability in half. Distributions assign chances to events so the axioms hold. Discrete cases use masses on points; continuous cases bury chance in intervals and densities.
A probability distribution says how chance is allotted among possible results of a random phenomenon—more precisely, among events, which are sets of outcomes. In plain talk it ranks which outcomes feel more probable; formally it is a probability measure that obeys the axioms. Random variables induce distributions on the values they take: a fair coin coded as one for heads and zero for tails puts probability one half on each of those numbers.
Sample spaces can be labels, numbers, or vectors—heads and tails for a flip, or one through six for a die. Discrete variables are often described by a probability mass function that loads each outcome. For a fair die, each face gets one sixth, so an even result sums to one half. Continuous variables on a continuum usually give individual points probability zero unless the density has wild spikes; only intervals and other rich sets carry positive chance.
Weighing supermarket ham to infinite precision makes exactly five hundred grams impossible in theory, yet a quality band from four hundred ninety to five hundred ten grams is a workable event. Continuous laws are often summarized by a cumulative distribution function—the chance the variable is at most some x—equal to the area under the density from minus infinity to x. Absolutely continuous cases also admit density functions whose small-interval chance is about f(x) times a tiny width.
Named families fill textbooks because they recur or matter theoretically. Whatever the description—mass, density, or cumulative—the assignment must stay nonnegative, at most one, and additive over countable collections of disjoint events. That Kolmogorov skeleton turns informal talk about likelihood into a measure on the right subsets of the sample space.
Source: Probability distribution
