# A Study of Assessment of Casinos’ Risk of Ruin in Casino Games with Poisson Distribution

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- We establish a formal basis for the risk of ruin through deriving the Poisson process, which contains the following factors: the win probability (p), the house advantage (a) and the predicted maximum number of bets (n).
- In addition, we provide a proof of the properties of the proposed formula, which satisfies monotonicity and also contains probabilistic features. This formula is used to illustrate the relevant cases in real casinos by setting changes of the factors in our proposed model.
- We explore and derive the experimental results of various models, and devise a novel measurement method with the risk of ruin scale, which aims at normalizing the risk levels and concretely reflecting the current risk statuses.

## 2. Literature Review

## 3. Deriving the Risk of Ruin

- p, representing the probability of a gambler winning in one round.
- ${p}^{n-k}$, representing the probability that a gambler always wins continuously from k to the goal n.

#### 3.1. Poisson Distribution for Risk of Ruin

#### 3.2. House Advantage Consideration

#### 3.3. The Property of Proposed $\mathit{Pro}\left(n\right)$

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

## 4. Empirical Simulation

#### 4.1. Data Analysis

^{−11}and 2.21 × 10

^{−12}, respectively, which implies that no one could bankrupt the casino even if the global population had participated in the game. Based on $n=\u230a\frac{\mathit{Bankroll}}{\mathit{MaxBet}}\u230b$, once the casino ensures its operating bankroll to be larger than $n\times \mathit{MaxBet}$, the risk of bankruptcy is negligible.

#### 4.2. Evaluation and Discussion

^{−2}. Compared to Kaufmann and Ralph-Vince, Coolidge’s results are more comprehensive and more readily identifiable. However, this method is still not perfect because the current number of wins/losses must be calculated before predictions can be made due to different outcomes being generated each time. Another limitation is that the upper bound of n is required to be defined, which is obviously not feasible in practice. Furthermore, the results produced by NBRM are similar to the proposed model, which is similar to Equation (7) except for taking the house advantage (a) into consideration. This is because they also make use of the binomial distribution in the derivation process: The NBRM focuses on the binomial distribution of a finite sample and the proposed model is conducted to describe the distribution from an infinite sample, thereby giving the probability of obtaining the entire event in the population. In addition, the ruin prediction can be further alleviated when considering the impact of profit ($a>0$). In fact, the simulated outcomes generated through these models are quite well understood and these models can roughly determine how each input feature influences the outcomes. We put forth the mathematical model that merely links up their relationship and describes how these changes affect the ROR through mathematical proof.

## 5. Risk of Ruin Scales

- The ROR measurement must be able to support high-precision decimals because it will receive the probability of ROR by $\mathit{Pro}\left(n\right)$.
- Minimum and maximum values should be determined in order to normalise the ROR measurement.
- Certainly, the maximum risk must be obtained when the input is 1.0.
- There exists a small deviation that can be negligible for approximating the limit of risk.

^{−10}as a sufficiently small value for delimiting. The input x is the predicted risk probability based on the proposed model. Since we are using a logarithmic scale that can represent the index of high-precision values, each step is spaced with a tenfold increase in probability. If $x=1.0$ is input, the result of Equation (11) will be equal to $10.0$, representing the maximum risk, and if $x\le S$, the result is less than or equal to $0.0$, representing the minimum/no risk.

`RUINED`status. In subsequent cases, when $n=20$, the risk of ruin is mitigated to $7.33$ (

`DANGEROUS`) and is further lowered to $4.92$ (

`RISKY`) when $n=40$. Finally, when $n>60$, the risk reaches

`RELAXED`status or lower. In summary, a casino can run its business free of worry of bankruptcy if the $\mathit{MaxBet}$ for each wager has also been configured to acceptably low levels.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Wager | p | a | $\mathit{n}=1$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Banker | 45.9% | 1.06% | 6.53 × 10^{−1} | 4.80 × 10^{−1} | 3.61 × 10^{−1} | 2.74 × 10^{−1} | 2.10 × 10^{−1} | 1.61 × 10^{−1} | 1.24 × 10^{−1} | 9.59 × 10^{−2} | 7.42 × 10^{−2} | 5.75 × 10^{−2} |

Player | 44.6% | 1.24% | 6.40 × 10^{−1} | 4.61 × 10^{−1} | 3.41 × 10^{−1} | 2.54 × 10^{−1} | 1.91 × 10^{−1} | 1.45 × 10^{−1} | 1.09 × 10^{−1} | 8.31 × 10^{−2} | 6.32 × 10^{−2} | 4.81 × 10^{−2} |

Tie | 9.5% | 14.36% | 1.27 × 10^{−1} | 2.00 × 10^{−2} | 3.26 × 10^{−3} | 5.43 × 10^{−4} | 9.18 × 10^{−5} | 1.57 × 10^{−5} | 2.74 × 10^{−6} | 4.85 × 10^{−7} | 8.74 × 10^{−8} | 1.61 × 10^{−8} |

Wager | p | a | $\mathit{n}=1$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Big/Small | 48.61% | 2.78% | 6.72 × 10^{−1} | 5.05 × 10^{−1} | 3.88 × 10^{−1} | 3.01 × 10^{−1} | 2.35 × 10^{−1} | 1.84 × 10^{−1} | 1.45 × 10^{−1} | 1.14 × 10^{−1} | 9.02 × 10^{−2} | 7.13 × 10^{−2} |

Dice Combinations | 13.90% | 2.8% | 2.38 × 10^{−1} | 6.84 × 10^{−2} | 2.04 × 10^{−2} | 6.17 × 10^{−3} | 1.88 × 10^{−3} | 5.77 × 10^{−4} | 1.77 × 10^{−4} | 5.47 × 10^{−5} | 1.69 × 10^{−5} | 5.22 × 10^{−6} |

Specific Doubles | 7.41% | 33.3% | 6.05 × 10^{−2} | 4.67 × 10^{−3} | 4.00 × 10^{−4} | 3.89 × 10^{−5} | 4.47 × 10^{−6} | 6.06 × 10^{−7} | 9.32 × 10^{−8} | 1.55 × 10^{−8} | 2.69 × 10^{−9} | 4.76 × 10^{−10} |

Any Triple | 2.78% | 30.6% | 1.84 × 10^{−2} | 4.35 × 10^{−4} | 1.15 × 10^{−5} | 3.58 × 10^{−7} | 1.44 × 10^{−8} | 7.49 × 10^{−10} | 4.61 × 10^{−11} | 3.08 × 10^{−12} | 2.13 × 10^{−13} | 1.49 × 10^{−14} |

Specific Triples | 0.46% | 30.1% | 1.82 × 10^{−3} | 4.24 × 10^{−6} | 1.12 × 10^{−8} | 4.05 × 10^{−11} | 2.57 × 10^{−13} | 2.44 × 10^{−15} | <1.0 × 10^{−15} | <1.0 × 10^{−15} | <1.0 × 10^{−15} | <1.0 × 10^{−15} |

Wager | p | a | $\mathit{n}=1$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Orange | 46.15% | 7.69% | 6.27 × 10^{−1} | 4.42 × 10^{−1} | 3.20 × 10^{−1} | 2.34 × 10^{−1} | 1.72 × 10^{−1} | 1.27 × 10^{−1} | 9.40 × 10^{−2} | 6.98 × 10^{−2} | 5.19 × 10^{−2} | 3.87 × 10^{−2} |

Purple | 25.08% | 7.69% | 3.78 × 10^{−1} | 1.68 × 10^{−1} | 7.76 × 10^{−2} | 3.63 × 10^{−2} | 1.71 × 10^{−2} | 8.08 × 10^{−3} | 3.84 × 10^{−3} | 1.83 × 10^{−3} | 8.75 × 10^{−4} | 4.19 × 10^{−4} |

Green | 15.38% | 7.69% | 2.39 × 10^{−1} | 6.91 × 10^{−2} | 2.07 × 10^{−2} | 6.28 × 10^{−3} | 1.92 × 10^{−3} | 5.93 × 10^{−4} | 1.84 × 10^{−4} | 5.70 × 10^{−5} | 1.78 × 10^{−5} | 5.54 × 10^{−6} |

Blue | 7.69% | 15.38% | 9.88 × 10^{−2} | 1.21 × 10^{−2} | 1.54 × 10^{−3} | 2.00 × 10^{−4} | 2.65 × 10^{−5} | 3.58 × 10^{−6} | 4.94 × 10^{−7} | 6.97 × 10^{−8} | 1.01 × 10^{−8} | 1.52 × 10^{−9} |

Yellow | 3.85% | 19.23% | 4.03 × 10^{−2} | 2.03 × 10^{−3} | 1.08 × 10^{−4} | 5.91 × 10^{−6} | 3.38 × 10^{−7} | 2.04 × 10^{−8} | 1.33 × 10^{−9} | 9.39 × 10^{−11} | 7.25 × 10^{−12} | 6.04 × 10^{−13} |

Logo 1/Logo 2 | 1.92% | 11.54% | 2.41 × 10^{−2} | 7.21 × 10^{−4} | 2.25 × 10^{−5} | 7.18 × 10^{−7} | 2.31 × 10^{−8} | 7.56 × 10^{−10} | 2.50 × 10^{−11} | 8.42 × 10^{−13} | 2.90 × 10^{−14} | <1.0 × 10^{−15} |

Wager | p | a | $\mathit{n}=1$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Red/Black | 18/37 | 2.7% | 6.73 × 10^{−1} | 5.06 × 10^{−1} | 3.89 × 10^{−1} | 3.02 × 10^{−1} | 2.36 × 10^{−1} | 1.85 × 10^{−1} | 1.46 × 10^{−1} | 1.15 × 10^{−1} | 9.09 × 10^{−2} | 7.20 × 10^{−2} |

Corner | 4/37 | 2.7% | 1.88 × 10^{−1} | 4.32 × 10^{−2} | 1.03 × 10^{−2} | 2.48 × 10^{−3} | 6.04 × 10^{−4} | 1.48 × 10^{−4} | 3.63 × 10^{−5} | 8.93 × 10^{−6} | 2.20 × 10^{−6} | 5.43 × 10^{−7} |

Split | 2/37 | 2.7% | 9.61 × 10^{−2} | 1.14 × 10^{−2} | 1.40 × 10^{−3} | 1.76 × 10^{−4} | 2.21 × 10^{−5} | 2.81 × 10^{−6} | 3.57 × 10^{−7} | 4.55 × 10^{−8} | 5.80 × 10^{−9} | 7.42 × 10^{−10} |

Single | 1/37 | 2.7% | 4.81 × 10^{−2} | 2.87 × 10^{−3} | 1.78 × 10^{−4} | 1.12 × 10^{−5} | 7.15 × 10^{−7} | 4.57 × 10^{−8} | 2.93 × 10^{−9} | 1.88 × 10^{−10} | 1.21 × 10^{−11} | 7.80 × 10^{−13} |

**Table 5.**ROR probabilities estimated by different models. Please refer to the above tables for variable configurations.

Game and Its Wager | Coolidge | Kaufmann | Ralph-Vince | NBRM | Equation (7) | Equation (8) | |
---|---|---|---|---|---|---|---|

Baccarat | Banker | 4.74 × 10^{−1} | 7.87 × 10^{−3} | 6.13 × 10^{−2} | 6.42 × 10^{−2} | 6.18 × 10^{−2} | 5.75 × 10^{−2} |

Player | 4.69 × 10^{−1} | 6.78 × 10^{−3} | 5.54 × 10^{−2} | 6.13 × 10^{−2} | 5.25 × 10^{−2} | 4.81 × 10^{−2} | |

Tie | 2.07 × 10^{−1} | 6.18 × 10^{−6} | 4.76 × 10^{−4} | 2.14 × 10^{−7} | 3.03 × 10^{−7} | 1.61 × 10^{−8} | |

Sic Bo | Big/Small | 4.32 × 10^{−1} | 1.06 × 10^{−2} | 7.52 × 10^{−2} | 6.81 × 10^{−2} | 8.46 × 10^{−2} | 7.13 × 10^{−2} |

Dice Combinations | 4.31 × 10^{−1} | 3.11 × 10^{−5} | 1.43 × 10^{−3} | 2.59 × 10^{−6} | 8.88 × 10^{−6} | 5.22 × 10^{−6} | |

Specific Doubles | 5.34 × 10^{−2} | 2.19 × 10^{−6} | 2.35 × 10^{−4} | 1.87 × 10^{−8} | 3.10 × 10^{−8} | 4.76 × 10^{−10} | |

Any Triple | 6.48 × 10^{−2} | 3.95 × 10^{−8} | 1.54 × 10^{−5} | 8.45 × 10^{−11} | 2.70 × 10^{−12} | 1.49 × 10^{−14} | |

Specific Triples | 6.71 × 10^{−2} | 2.82 × 10^{−11} | 1.12 × 10^{−7} | 5.09 × 10^{−12} | <1.0 × 10^{−15} | <1.0 × 10^{−15} | |

Big Six | Orange | 3.23 × 10^{−1} | 8.10 × 10^{−3} | 6.25 × 10^{−2} | 6.13 × 10^{−2} | 6.37 × 10^{−2} | 3.87 × 10^{−2} |

Purple | 3.23 × 10^{−1} | 4.23 × 10^{−4} | 8.41 × 10^{−3} | 1.82 × 10^{−3} | 1.09 × 10^{−3} | 4.19 × 10^{−4} | |

Green | 3.23 × 10^{−1} | 4.82 × 10^{−5} | 1.92 × 10^{−3} | 2.91 × 10^{−5} | 2.11 × 10^{−5} | 5.54 × 10^{−6} | |

Blue | 1.93 × 10^{−1} | 2.56 × 10^{−6} | 2.61 × 10^{−4} | 1.82 × 10^{−9} | 4.37 × 10^{−8} | 1.52 × 10^{−9} | |

Yellow | 1.47 × 10^{−1} | 1.48 × 10^{−7} | 3.78 × 10^{−5} | 1.07 × 10^{−12} | 6.30 × 10^{−11} | 6.04 × 10^{−13} | |

Logo 1/Logo 2 | 2.51 × 10^{−1} | 8.83 × 10^{−9} | 5.55 × 10^{−6} | 4.94 × 10^{−13} | 7.26 × 10^{−14} | <1.0 × 10^{−15} | |

Roulette | Red/Black | 4.34 × 10^{−1} | 1.07 × 10^{−2} | 7.54 × 10^{−2} | 6.63 × 10^{−2} | 8.50 × 10^{−2} | 7.20 × 10^{−2} |

Corner | 4.34 × 10^{−1} | 1.07 × 10^{−5} | 6.90 × 10^{−4} | 2.22 × 10^{−8} | 9.72 × 10^{−7} | 5.43 × 10^{−7} | |

Split | 4.34 × 10^{−1} | 5.95 × 10^{−7} | 9.71 × 10^{−5} | 1.55 × 10^{−10} | 1.61 × 10^{−9} | 7.42 × 10^{−10} | |

Single | 4.34 × 10^{−1} | 3.52 × 10^{−8} | 1.42 × 10^{−5} | 8.65 × 10^{−13} | 2.05 × 10^{−12} | 7.80 × 10^{−13} |

Scale | Status | One Out of the Number of People That Can Bankrupt the Casino by the Same Game |
---|---|---|

$\le \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}1.25$ | SAFE | The casino cannot be bankrupted even against the world’s population |

$\left(1.25,2.50\right]$ | RELAXED | Approx. 600 million people (half the size of China’s population) |

$\left(2.50,3.75\right]$ | RISKY | Approx. 30 million people (the size of a small- to mid-sized province in China in terms of population) |

$\left(3.75,5.00\right]$ | Approx. 1,750,000 people (the size of a small- to mid-sized city in terms of population) | |

$\left(5.00,6.25\right]$ | DANGEROUS | Approx. 100,000 people |

$\left(6.25,7.50\right]$ | Approx. 5000 people | |

$\left(7.50,8.75\right]$ | RUINED | Everyone could bankrupt the casino |

$\left(8.75,10.0\right]$ | Casinos can be considered bankrupt |

Wager | p | a | $\mathit{n}=1$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Orange | 46.15% | 7.69% | 9.80 | 9.65 | 9.51 | 9.37 | 9.24 | 9.10 | 8.97 | 8.84 | 8.72 | 8.59 |

Purple | 25.08% | 7.69% | 9.58 | 9.23 | 8.89 | 8.56 | 8.23 | 7.91 | 7.58 | 7.26 | 6.94 | 6.62 |

Green | 15.38% | 7.69% | 9.38 | 8.84 | 8.32 | 7.80 | 7.28 | 6.77 | 6.26 | 5.76 | 5.25 | 4.74 |

Blue | 7.69% | 15.38% | 8.99 | 8.08 | 7.19 | 6.30 | 5.42 | 4.55 | 3.69 | 2.84 | 2.00 | 1.18 |

Yellow | 3.85% | 19.23% | 8.61 | 7.31 | 6.03 | 4.77 | 3.53 | 2.31 | 1.12 | <0.0 | <0.0 | <0.0 |

Logo 1/Logo 2 | 1.92% | 11.54% | 8.38 | 6.86 | 5.35 | 3.86 | 2.36 | 0.88 | <0.0 | <0.0 | <0.0 | <0.0 |

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**MDPI and ACS Style**

Siu, K.-M.; Chan, K.-H.; Im, S.-K.
A Study of Assessment of Casinos’ Risk of Ruin in Casino Games with Poisson Distribution. *Mathematics* **2023**, *11*, 1736.
https://doi.org/10.3390/math11071736

**AMA Style**

Siu K-M, Chan K-H, Im S-K.
A Study of Assessment of Casinos’ Risk of Ruin in Casino Games with Poisson Distribution. *Mathematics*. 2023; 11(7):1736.
https://doi.org/10.3390/math11071736

**Chicago/Turabian Style**

Siu, Ka-Meng, Ka-Hou Chan, and Sio-Kei Im.
2023. "A Study of Assessment of Casinos’ Risk of Ruin in Casino Games with Poisson Distribution" *Mathematics* 11, no. 7: 1736.
https://doi.org/10.3390/math11071736