A366566 a(n) is the expected end time of a game with three gamblers, one of which starts with capital n, the others with capital 1 each. The end time, rounded to the nearest integer, is given for games in which one of the two poor players wins.

3, 6, 9, 13, 17, 22, 28, 34, 41, 49, 58, 67, 76, 87, 98, 109, 122, 135, 149, 163, 178, 194, 210, 227, 245, 263, 282, 302, 322, 343, 365, 387, 410, 434, 458, 483, 509, 535, 562, 590, 619, 648, 677, 708, 739, 770, 803, 836, 869, 904, 939, 974, 1011, 1048, 1085

Offset: 1

Author

Hugo Pfoertner, Oct 13 2023

Keywords

• nonn

Comments

For details see the Diaconis link.

Initially, terms up to a(25) were calculated using Monte Carlo simulation of 10^9 games at each value of n.

The expected end times without rounding to nearest integer are: 3.00, 5.57, 8.76, 12.57, 17.03, 22.14, 27.91, 34.33, 41.41, 49.15, 57.55, 66.61, 76.33, 86.72, ... .

The expected shorter end time also allowing the rich player to win would be 2*n+1 (Bachelier, 1912, page 149).

Links

• Louis Bachelier, Calcul des probabilités. Tome I, Gauthier-Villars, Paris, 1912.
• Persi Diaconis and Stewart N. Ethier, Gambler’s Ruin and the ICM, Statist. Sci. 37 (3) 289 - 305, August 2022.
• Persi Diaconis, Gambler's ruin with k gamblers (slide 3), talk in the Rutgers Experimental Mathematics Seminar, Fall 2023 Semester, Oct. 12, 2023.
• Experimental Mathematics, GAMBLER’S RUIN WITH K GAMBLERS, recording of talk, Vimeo video (time after 11:55), Oct 22, 2023.
• Hugo Pfoertner, Example of the time history of a game with n=3, i.e., the "rich" player starts with 3 chips.
• Hugo Pfoertner, Distribution of the number of games won, n=3, plotted vs end time.
• Hugo Pfoertner, Distribution of the number of games won, n=5, plotted vs end time.
• Hugo Pfoertner, Distribution of the number of games won, n=6, plotted vs end time.
• Hugo Pfoertner, Distribution of the number of games won, n=10, plotted vs end time.
• Hugo Pfoertner, Distribution of the number of games won, n=20, plotted vs end time.

Crossrefs

Cf. A366166, A366567 (mode of corresponding probability distributions), A366995, A366996.

Formula

a(n) equals A366995(n)/A366996(n) rounded to the nearest integer. - Pontus von Brömssen, Oct 31 2023

Extensions

a(26)-a(55) from Pontus von Brömssen, Oct 31 2023