Searching for viable betting strategies in gambling, with consideration to Roulette
Joseph Toninato UMD Honors Capstone Project Introduction
Gambling is over
8000 years old
1.6 billion people
gambled last year
Mostly in casinos Roulette
Wheel with 38 slots, 37 in French version
Slots numbered 0-36, and 00
Ball is spun while wheel turns
Lands in slot
Multitude of betting options
House advantage due to 0 and 00
Is there a way to beat the system? Strategies
Betting factor k = 1/p, where p is probability of winning
W(n) is win function
L(n) is loss function
P(n) is profit function
Initial bet was always $1
No house advantage Martingale System
Double bet after each loss
Only bet on odds of 1:2
n W(n) = 2Bn, where the bet at turn n Bn = 2 B0
L(n) = Bn – B0
P(n) = B0 Other studies have found it to be ineffective Constant Bet
Bet $1 every turn
W(n) = k
L(n) = n
P(n) = k – n
E[P] = n(k-1)n/ (k-1) ~ approaches 0 as n increases Modified Martingale
Instead of doubling with odds of 1:2, multiply next bet by k for odds 1:k
W(n) = kn
L(n) = (kn-1)/(k-1)
P(n) = kn - (kn-1)/(k-1)
E[P] = (k-1)n-1 – (1/kn)
Experimental
Similar to the previous, but multiply bet by k only after k losses in a row.
Experimental Additional Strategies
For each below, switch happens when player cannot make next necessary bet
Start with experimental, switch to constant bet
Start with experimental, switch to modified Martingale
Start with modified Martingale, switch to constant bet Methods
Computer simulations where random number generator acted as wheel Turn Based: Allow system to work out until round 100 or player is out of money Win In One: Trial stops after player wins once
Manipulated strategy used, starting money S, and k.
S = 100, 1000, and unlimited
Recorded turns, wins, money for 1000 trials
Found mean, standard deviation, median, and proportion of positive profits Results
No strategy allowed for perfect beating of the system
High occurrence in positive profits coincided with high money input
Modified Martingale system with S being unlimited produced high chance of winning, and high median profit
Future Work
Test other strategies in similar matter
Find how much money is needed to have favorable odds
Alter switching strategies to better suit player
References
History in an Hour. (2012). The Ancient Ages of Gambling. Retrieved from http://www.historyinanhour.com/2012/10/22/the- ancient-ages-of-gambling/
AddictionBlog. (2011). How many people gamble? Retrieved from http://gambling.addictionblog.org/how-many-people-gamble/
Biography. (2015) Blaise Pascal Biography. Retrieved from http:// www.biography.com/people/blaise-pascal-9434176
Turner N. (1998). Doubling vs Constant Bets as Strategies for Gambling. Jounral of Gambling Studies 14, 413-429
Snell JL. (1982) . Gambling, Probability and Martingales. The Mathematical Intelligencer 4, 118-124
Neal DK, Russell MD. (2009). A Generalized Martingale Betting Strategy. Missouri Journal of Mathematical Science 21, 183-197