A395267 - OEIS

A395267

Decimal expansion of 1/3 + 3*sqrt(3)/(4*Pi).

7, 4, 6, 8, 3, 0, 0, 0, 4, 8, 9, 9, 6, 7, 7, 3, 7, 0, 4, 6, 6, 8, 2, 8, 2, 0, 7, 0, 6, 8, 0, 6, 0, 4, 1, 4, 3, 8, 1, 3, 7, 3, 1, 9, 3, 6, 0, 8, 3, 1, 3, 8, 2, 2, 9, 3, 2, 8, 5, 7, 8, 4, 8, 5, 5, 1, 6, 0, 6, 0, 6, 0, 9, 3, 5, 2, 8, 5, 6, 7, 9, 4, 6, 3, 6, 1, 8, 3, 5, 4, 9, 5, 0, 7, 1, 1, 5, 0, 0, 2, 3, 9, 1, 0, 6

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OFFSET

0,1

COMMENTS

The probability that a chord determined by two points independently and uniformly selected at random in a disk is longer than a side of an equilateral triangle inscribed in the disk (Garwood and Holroyd, 1966).

Bertrand (1889) gave three possible definitions for a random chord, for which the corresponding probabilities are 1/3, 1/2, or 1/4.

Note that the probability calculated by Garwood and Holroyd (1966) is that the chord's distance from the center of the disk is larger than half the radius. It equals 1 minus this constant, i.e., 2/3 - 3*sqrt(3)/(4*Pi) = 0.253169... .

REFERENCES

Gábor J. Székely, Paradoxes in Probability Theory and Mathematical Statistics, Reidel, 1986, p. 47.

LINKS

Table of n, a(n) for n=0..104.

Vangalur S. Alagar and Larry H. Thiel, Algorithms for detecting M-dimensional objects in N-dimensional spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol PAMI-3, No. 3 (1981), pp. 245-256.

Joseph Bertrand, Calcul des probabilités, Gauthier-Villars, Paris, 1889, pp. 5-6.

F. Garwood and E. M. Holroyd, The Distance of a "Random Chord" of a Circle from the Centre, The Mathematical Gazette, Vol. 50, No. 373 (1966), pp. 283-286.

P. A. P. Moran, A second note on recent research in geometrical probability, Advances in Applied Probability , Vol. 1, No. 1 (1969) , pp. 73-89.

Eric Weisstein's World of Mathematics, Bertrand's Problem.

Wikipedia, Bertrand paradox (probability).

Index entries for transcendental numbers.

EXAMPLE

0.746830004899677370466828207068060414381373193608313...

MATHEMATICA

RealDigits[1/3 + 3*Sqrt[3]/(4*Pi), 10, 120][[1]]

PROG

(PARI) 1/3 + 3*sqrt(3)/(4*Pi)

CROSSREFS

Cf. A010701, A020761, A020773, A395268.