OFFSET
1,8
LINKS
Louis Bachelier, Calcul des probabilités. Tome I, Gauthier-Villars, Paris, 1912.
Eric Weisstein's World of Mathematics, Random Walk--1-Dimensional.
Wikipedia, One-dimensional random walk
FORMULA
P(m,mue) = (mue/m) * mue! / (((mue - m)/2)! * ((mue + m)/2)!) * (1/2)^mue, with round count mue and initial capital m (Bachelier, 1912, page 103).
P(m,mue) = 0 for mue < m and for mue + m odd.
EXAMPLE
The triangle begins
1;
1, 1;
1, 1, 1;
1, 3, 5, 5;
1, 1, 9, 7, 7;
1, 5, 7, 7, 21, 21;
1, 3, 5, 3, 45, 33, 33;
The numbering is as follows, with
[n, rounds, P] = [initial capital, coin flips, probability of ruin]:
[1, 1, 1/2]
[2, 2, 1/4] [1, 3, 1/8]
[3, 3, 1/8] [2, 4, 1/8] [1, 5, 1/16]
[4, 4, 1/16] [3, 5, 3/32] [2, 6, 5/64] [1, 7, 5/128]
[5, 5, 1/32] [4, 6, 1/16] [3, 7, 9/128] [2, 8, 7/128] [1, 9, 7/256]
[6, 6, 1/64] [5, 7, 5/128] [4, 8, 7/128] [3, 9, 7/128] [2, 10, 21/512]
[1, 11, 21/1024]
With initial capital odd, ruin can only occur at odd numbered rounds, with even initial capital only at even numbered rounds.
CROSSREFS
KEYWORD
AUTHOR
Hugo Pfoertner, Oct 24 2023
STATUS
approved