geometric probability

Geometric probability is the continuous cousin of the classical definition. When a point is chosen 'at random and uniformly' from a region — a segment of line, an area, or a volume — the probability that it lands in some part of that region is the size of the part divided by the size of the whole, where size means length, area, or volume as appropriate. Throw a dart that is equally likely to land anywhere on a board, and the chance it hits the bullseye is the bullseye's area divided by the board's area.

It is the right tool whenever the sample space is a continuum, because there you cannot count outcomes — there are infinitely many points, and each single point has probability zero. So instead of counting, you measure. To find P(the random point falls in set A), compute (measure of A) / (measure of the whole region). On a line you divide lengths; in a plane you divide areas; the recipe is the classical favourable-over-total idea with measure standing in for a count. The word 'uniformly' is essential — it is the continuous version of 'equally likely', meaning no region is favoured over another of the same size.

Two honest points. First, that each individual point has probability zero is not a paradox: probability now lives in regions of positive size, not in single points, and 'probability zero' here means 'negligible extent', not 'impossible'. Second, geometric probability is famous for traps when 'at random' is not pinned down — the Bertrand paradox shows that 'a random chord of a circle' can give different answers depending on exactly how you randomize, a warning that you must specify your random mechanism precisely before any number is meaningful.