Concept

Geometric probability

Probability computed from lengths, areas or volumes, when points, lines or arcs are placed at random in a region. It turns a question of chance into one of measuring a shape, and its classic results include Buffon's needle and the chance that random arcs cover a circle.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Random points and the hull around them. triangle: 13 hull corners of 400 points; square: 17 hull corners of 400 points; disc: 22 hull corners of 400 points.

The corners of a random hull

Drop ten thousand points at random into a triangle and stretch a band round them: about twenty points touch it. Drop them into a disc and about seventy-three do. The count grows like a logarithm in one shape and like a cube root in the other, and the reason is where the boundary bends.

analysis · Convexity

Arcs that overlap and still leave gaps. n=14, a=0.2: starts 0.014,0.017,0.118,0.192,0.232,0.237,0.251,0.428,0.458,0.500,0.510,0.579,0.930,0.931; gaps 1; P(cover) exact 0.34675223.

Arcs that cover a circle

Throw fourteen arcs, each a fifth of a circle, onto the circle at random: together they are nearly three times as long as it, and still a third of the time some stretch is left bare. The chance of covering is an alternating sum over which gaps are too long, found by W. L. Stevens in 1939. Its terms run into the thousands and cancel to almost nothing, and they say that random covering costs a logarithm.

probability · Inclusion exclusion

Named alongside it

The objects these essays reach for when they reach for this one.

Asymptotics Bonferroni inequalities Convex hull Convexity Coupon collector Exact arithmetic Expected value Gumbel distribution Inclusion exclusion Logarithm Symmetry Uniform distribution

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