Kyler Lindo 10/4/23

Rolling The Odds: Unraveling Dice Probability

Abstract

At the beginning of this study, there were questions revolving around whether rolling D6 dice (six-sided dice) can become predictable with repetition. When rolling dice it’s important to realize that each toss is independent and distributed between integers of one – six. The differing results obtained give certainty to how independent each dice roll truly is. Along with an intriguing trend associated with the sum of both die values. However, these computations lacked a certain magnitude over predicting what numbers would be rolled.

Introduction

Does increasing the number of trial D6 dice rolled equate to increased sum predictability? The independent outcome of each dice toss is what makes the probability of prediction more like a guessing game. This experiment is vital to understanding the possible array of outcomes displayed by documented trials to give a concise answer on D6 dice predictability. In Pascal and Fermat Dice Probability by Elizabeth M. Glaister and Paul Glaister, a few of their studies consisted of the sum of numbers D6 dice provided with each trial. It was seen that in some trials the total sum of the dice rolled would equal each other, P(a) = P(b). Even so, at times when the dice were rolled the sum of the P(b) would be greater than P(a). This resulted in a higher probability that P(b) was going to result in higher values. It’s important to notice that with both trials there were varying possibilities on which die would come out on top with the greater sum. The research that I performed extends that idea with the use of an individual die-rolling automated system. Along with confirming the varying summed possibilities in Paul and Elizabeth’s lab report.

Materials

An online automated die/dice rolling simulator
Graph formatting software
Equation formatting software
Data Box
Google Sheets

Methods

This particular lab experiment was conducted within a wifi available location. A website was used to perform the rolling of die/dice. After every automated roll, the given information was recorded in an online data sheet. To keep the data sheet organized and signify the result of each trial there were title descriptions placed directly above. Some difficulties were encountered trying to retain the percent value of each roll probability. I later found that to effectively keep track of the values the math equation, [roll_probability_percent = (Number of Outcomes / Total Number of Trials) * 100], was put in place. This was done by formatting each cell to calculate specific values based on the information entered. Once the data within the cells were complete and accurate a chart was formed to better exemplify the data results signifying the completion of the procedure.

Results

The probability of rolling a specific integer for Die #1 is displayed in Figure 1. The probability of rolling a specific integer for Die #2 is displayed in Figure 2. The sum of these integers and outcome probability is displayed in Figure 3.

Figure 1 Figure 2

Figure 3

Discussion

A quantitative data table was created to display the percent value of times a specific integer showed up when the dice were rolled for one hundred trials. This was done to completely support Paul and Elizabeth’s understanding of how unpredictable the values to be rolled truly are. In comparison to their lab experiment, varying probabilities of differing particular values were apparent. For example, the Die #1 chart in the data shows that it’s most likely to roll a numerical value of six for a one-hundred-roll trial. Despite those odds, Die #2 chart values obtained intriguingly deny any form or predictability of any certain individual numerical outcome. For instance, in the data displayed under Probability of Rolling Numerical Value for Die #2, values one and two have the exact 19% chance of being rolled within a one-hundred-roll trial. Along with values three and four which have the exact 14% chance of being rolled. There isn’t much predictability displayed in that scenario. I also took my experiment further to dispute theoretical beliefs with experimental values. It is assumed that within a 108-roll trial, the sum of both dies has a higher chance of equating to a value of seven. Statistically, there are six ways to obtain the summed value of seven when rolling D6 dice. This is deemed to hold a higher possibility than any other value. In my report, rolling a sum of eight was more probable than rolling a sum of seven by a fairly large margin.

Conclusion

The significance of these trials is aimed to provide better insight into experimental values on D6 dice probability. As well as argue against values expected by theoretical computations. The data obtained was to exemplify how increasing the number of times D6 dice are rolled doesn’t equate to increased sum predictability. Enough odds are rolling one die, now imagine those odds multiplied by rolling 2. The odds of predicting a specific outcome will always be the limiting factor.

Citation

Glaister, E. M., & Glaister, P. (2012). Pascal and Fermat Dice with Probability. Mathematics in School, 41(3), 28–29. http://www.jstor.org/stable/23269223

Pratt, D. (2000). Making Sense of the Total of Two Dice. Journal for Research in Mathematics Education, 31(5), 602–625. https://doi.org/10.2307/749889

Appendix [A]

Values rolled for each Die#1 and Die #2, and the sum of values rolled for Die #1 and Die #2

LAB REPORT