# Journal of Mathematics and Statistics

## An Investigation into Labouchère’s Betting System to Improve Odds of Favorable Outcomes to Generate a Positive Externality Empirically

Jake Billings and Sebastian Del Barco

DOI : 10.3844/jmssp.2017.186.196

Journal of Mathematics and Statistics

Volume 13, 2017

Pages 186-196

### Abstract

The Labouchère gambling system is hypothesized to increase the probability of winning a predetermined arbitrary profit in a gambling system such as a coin flip or a roulette game in which both payouts and odds are 1:1. However, use of the system increases the downside monetary risk in the event of a streak of multiple losses. To begin, a player creates an arbitrary series of consecutive integers with a sum equal to the desired profit from multiple rounds of betting. Using the system, a player will either win an amount equal to the sum of the elements of the initial series or lose all of their available capital. This sequence was simulated multiple times to determine the statistical characteristics of both the return and of the loss in an average round of betting. By running the simulations of millions of rounds of Labouchere, it was possible to discern the probable outcomes of running the system using the Labouche gambling sequence and plotting the results on a graph to map the average return on the initial capital investment. The Labouchère system is very psychologically appealing to players because when applied over time it provides very consistent linear returns. However, there is eventually a critical moment at which the available capital for betting is exceeded and a player loses all of their available capital. It was found that as the number of bets increased, the outcome of applying the sequence approached zero.