Video Poker: Bankroll Size vs. Risk of Ruin

Introduction

This appendix addresses the question of bankroll size Vs. risk of ruin in video poker. For those who don't know, the risk of ruin is the probability of losing an entire bankroll. The following tables show the number of betting units required according to the desired risk of ruin, the game, and cash back. A "betting unit" is five coins, for example a betting unit would be $1.25 for a 25 cent machine player.

As an example the full play deuces wild player, with 0.25% cash back, would need a bankroll of 3333 units to have a probability of ruin of 5%. See the following chart to find this number. These numbers may seem high compared to other sources based on ruin before some other event is achieved. The tables below are for ruin at any time over an infinite period of time and thus have no successful terminating event, other than reaching an infinite bankroll. Consequently these tables are best used by the player considering establishing a bankroll for an indefinite period of play

 

The following table applies to "full pay" deuces wild. This pay table can be found in my video poker tables but is generally marked by paying 5 for a four of a kind. The expected return on this game is 100.76% and the standard deviation is 5.08.

 

 

The following table applies to "10/7" double bonus. This pay table can be found in my video poker tables but is generally marked by paying 7 for a flush and 10 for a full house. The expected return on this game is 100.17% and the standard deviation is 5.32.

 

 

The following table applies to "full pay" jacks or better. This pay table can be found in my video poker tables but is generally marked by paying 6 for a flush and 9 for a full house. The expected return on this game is 99.54% and the standard deviation is 4.42.

 

 

Methodology

An entirely mathematical approach was used to create the above tables. The theory was similar to that of the solution of problem 72 in my site of math problems. Briefly if p is the probability of ruin with 1 unit then p² is the probability of ruin with 2 units, p³ is the probability of ruin with 3 units, and so on. With the known probabilities for the outcome of each hand an equation could be set up to solve: p=sum over all possible outcomes of pr_i * p^r_i, where pr_i is the probability of hand i and r_i is the return for hand i. Using an iterative process I solved for p. The cash back was given to the player at every hand. For example if the cash back rate was 1% than one penny was added to each win, including no win at all, for each $1 bet.
 

 

Written by: Michael Shackleford