发表机构
BYU–Hawaii（杨百翰大学夏威夷分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对非公平的五棱柱骰子，结合其几何特性与4.1万次投掷实验数据，构建了含两个调整参数的统计概率模型，模型在测试的高径比范围内拟合良好。
AI 中文摘要
在概率论与游戏设计领域，骰子概率的研究至关重要。常见骰子通常是公平的，无需投掷即可预测其概率，但五棱柱骰子并不公平，因此其概率更为复杂。本研究的数学模型自然源自骰子的几何特性与经验证据，设计了体积恒定但尺寸不同的骰子，采用填充率100%的3D打印机制作骰子并准备就绪，使用模拟骰子通过骰子塔掉落至坚硬表面的机器获取了约41000次投掷结果。基于这些数据构建了统计模型，用于计算不同高径比骰子的概率，该模型公式基于质心立体角，但因不同静止面之间转换的能量及其他要求，引入了两个调整参数。在测试的高径比范围内，实验数据与模型的偏差处于合理统计误差内。
英文摘要
In the realm of probability theory and game design, the study of dice probabilities plays a crucial role. The featured dice are usually fair, so their probabilities can be predicted without rolling the dice. Pentagonal prismatic dice are not fair. Hence, the probabilities are more complicated. Our mathematical model is naturally derived from both the geometric properties of the dice and empirical evidence. Dice of varying sizes but with constant volume were designed. Using a 3D printer with 100 percent infill, the dice were printed and prepared. A machine that simulates dropping dice through a dice tower onto a hard surface was used to obtain about 41,000 results. From these data, a statistical model was constructed that gives probabilities of dice with varying height-to-radius ratios. The formula for the model is based on the centroid solid angle, but with two adjustment parameters due to varying energy and other requirements to transition between different resting aspects. Within the range of height-to-radius ratios tested, the data are within reasonable statistical error of the model.
Comments10 pages, 8 figures
