Rolling the Dice:
Questioning the Probabilities
Shadman Azad
Abstract:
This experiment aimed to explore the probability distribution of sums obtained from rolling a pair of dice. A total of 500 rolls were conducted, and the sums of the dice were recorded and analyzed. The results were compared to theoretical probabilities and a peer-reviewed study on dice probability. The data revealed that the empirical distribution closely matched the expected theoretical distribution, with slight deviations attributable to random variation. The study highlights the practical application of probability theory in predicting outcomes of random events and underscores the importance of sample size in empirical experiments.
Introduction
Have you ever wondered why rolling a 7 in dice games happens more often than rolling a 2 or a 12? Probability theory provides the answer, revealing that not all outcomes are equally likely when rolling a pair of dice. Dice have been used for centuries in games, decision-making, and even divination, making them a fascinating subject for studying randomness and probability. This experiment was designed to explore the empirical probability distribution of dice sums and compare it to the theoretical expectations. By rolling a pair of dice 500 times, we aimed to observe how closely real-world results align with mathematical predictions. The hypothesis is that the results and empirical probabilities would closely match the theoretical probabilities, with some deviation due to randomness, because the theoretical probability shows that not all outcomes are equally likely.
Materials and Methods
Materials:
- A computer or mobile device with internet access
- The website https://rolladie.net/ for simulating dice rolls
Procedure:
- Open the website https://rolladie.net/ on your device.
- Set the website to roll 2 dice (6-sided each) simultaneously.
- Configure the website to roll the dice 500 times.
- Record the sum of the two dice in a table for each roll.
- After completing all rolls, tally the frequency of each sum (2 through 12).
- Calculate the empirical probability for each sum by dividing its frequency by the total number of rolls (500).
- Compare the results to the theoretical probabilities and organize the data into a table and create a bar chart to visually compare the empirical and theoretical distributions.
Results
The results of the experiment are presented below, showcasing the frequency and probability of each sum obtained from rolling a pair of dice 500 times. The data is organized into a table and a bar graph, as shown below:
Fig. 1: Frequency and Probability of Each Sum
| Sum | Frequency | Probability (freq/500) | Theoretical Probability |
| 2 | 14 | 0.028 | 0.028 |
| 3 | 28 | 0.056 | 0.056 |
| 4 | 42 | 0.084 | 0.083 |
| 5 | 56 | 0.112 | 0.111 |
| 6 | 69 | 0.138 | 0.139 |
| 7 | 83 | 0.166 | 0.167 |
| 8 | 72 | 0.144 | 0.139 |
| 9 | 55 | 0.110 | 0.111 |
| 10 | 42 | 0.084 | 0.083 |
| 11 | 27 | 0.054 | 0.056 |
| 12 | 12 | 0.024 | 0.028 |
This table shows the frequency and probability of each sum (2 through 12) obtained from 500 dice rolls. The empirical probabilities closely align with the theoretical probabilities (Grinstead & Snell, 2012, Chapter 1, Section 1.2), with minor deviations.
Fig. 2: Empirical vs. Theoretical Probabilities
This bar graph compares the empirical probabilities (observed from the experiment) with the theoretical probabilities (calculated mathematically). The results show a close alignment between the two, with slight variations due to random chance.
Data Analysis
The results of the experiment closely aligned with the hypothesis, which predicted that the empirical probabilities of dice sums would match the theoretical probabilities, with minor deviations due to random variation. As shown in Table 1 and Fig. 1, the empirical probabilities for each sum (2 through 12) were very close to the theoretical values. For example, the sum of 7 occurred 83 times (empirical probability of 0.166), compared to the theoretical probability of 0.167. Similarly, the sum of 2 occurred 14 times (empirical probability of 0.028), matching the theoretical probability of 0.028. These findings confirm that the hypothesis was correct, and the experiment successfully demonstrated the reliability of probability theory in predicting outcomes of random events. The minor deviations observed, such as the sum of 8 occurring slightly more often than expected, can be attributed to the inherent randomness of the experiment and the limited sample size.
The results of this experiment are consistent with the findings of Glaister and Glaister (2012), who explored the probability of outcomes when rolling non-standard dice pairs. In their study, they analyzed the likelihood of one die outperforming another based on the numbers on their faces, demonstrating that even small changes in the distribution of numbers can influence outcomes. While their study focused on non-standard dice, the underlying principles of probability align with the results of this experiment. Both studies highlight the importance of combinatorial mathematics in predicting outcomes and emphasize that empirical results, when conducted systematically, tend to align with theoretical expectations. This consistency across studies reinforces the validity of probability theory in analyzing random events.
Conclusion
The experiment investigated the probability distribution of the sum of two dice rolled in 500 trials. The results demonstrated that theoretical probabilities closely aligned with the observed probabilities with slight differences, which could be attributed to randomness. The most occurring sums of 7 and the least occurring sums of 2 and 12 were found, as probability theory predicted. These results support the validity of theoretical probability, which states predicting outcomes of random events and highlight the importance of sample size in empirical experiments.
The results of this study have important practical applications in many fields, such as statistics, game design, and risk analysis, where a good understanding of probability distributions is essential. For example, game designers can use these findings to create balanced game mechanics, while statisticians can utilize the principles to model real-life random events. Future experiments could examine the effects of larger sample sizes, weighted dice, or non-traditional dice (e.g., 10-sided or 20-sided). Additionally, studies could examine the effect of variations in dice shape or rolling technique on results, building on the work performed by Glaister and Glaister (2012) in their study.
Works Cited
Diaconis, P., & Keller, J. B. (1989). Fair dice. The American Mathematical Monthly, 96(4), 337–339. https://doi.org/10.2307/2324095
Glaister, E. M., & Glaister, P. (2012). Pascal and Fermat dice with probability. Mathematics in School, 41(3), 28–29. https://www.jstor.org/stable/23269223
Grinstead, C. M., & Snell, J. L. (2012). Introduction to probability (2nd ed.). American Mathematical Society. Retrieved from https://math.dartmouth.edu/~prob/prob/prob.pdf
Appendix
Appendix A: Website Used for Dice Rolls
The experiment was conducted using the online dice-rolling tool available at:
Appendix B: Theoretical Probability Calculations
| Sum | Number of Combinations | Theoretical Probability |
| 2 | 1 | 1/36 ≈ 0.028 |
| 3 | 2 | 2/36 ≈ 0.056 |
| 4 | 3 | 3/36 ≈ 0.083 |
| 5 | 4 | 4/36 ≈ 0.111 |
| 6 | 5 | 5/36 ≈ 0.139 |
| 7 | 6 | 6/36 ≈ 0.167 |
| 8 | 7 | 5/36 ≈ 0.139 |
| 9 | 8 | 4/36 ≈ 0.111 |
| 10 | 9 | 3/36 ≈ 0.083 |
| 11 | 10 | 2/36 ≈ 0.056 |
| 12 | 11 | 1/36 ≈ 0.028 |
Appendix C: References to Academic Studies
Grinstead & Snell (2012): Theoretical probabilities were derived from Introduction to Probability (Chapter 1, Section 1.2).
Glaister & Glaister (2012): Comparative analysis was based on their study of non-standard dice probabilities.
