The largest gap between points on a circle

Question

n points placed independently and uniformly on a circle of circumference 1 divide it into n arcs (gaps). What is the probability that the largest gap exceeds a fraction g of the circle? Evaluate n = 4, g = 	frac13.

Accepted Answer

The probability that one specified gap exceeds g is (1 - g)^{\,n-1}: fixing a gap of length more than g leaves the remaining points to fall in a shortened circle. More generally, the probability that k specified gaps all exceed g is (1 - kg)^{\,n-1} when kg \le 1, and 0 otherwise (you cannot fit k disjoint arcs longer than g if they would overrun the circle). By inclusion-exclusion over which gaps are large, P(	ext{largest gap} g) = \sum{k=1}^{\lfloor 1/g floor} (-1)^{k+1}\binom{n}{k}\,(1 - kg)^{\,n-1}. For n = 4, g = 	frac13 (so terms run while kg \le 1, i.e. k \le 3): - k = 1: \binom{4}…

The largest gap between points on a circle

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