**Dice Sum Probability Analysis:** **Exploring Sum Distributions in Dice Rolls**

Kevin Wang

13 March, 2024

**Abstract**

This dice probability experiment explores the cumulative outcomes of rolling a pair of six-sided dice 100 times. The aim of the experiment is to record and analyze the sums obtained from each roll, in order to learn the concept of probability, and to find patterns associated with it. To do the experiment, a website named rolladie.net that randomly generates dice numbers was used to roll two digital dice 100 times. The data collected showed that among the 100 rolls, the sum of 7 appeared the most frequent, while the sum of 12 was the least. By examining these outcomes, it helps us understand the concept of probability, and patterns that exist in the real world.

**Introduction**

This experiment involves digitally rolling a pair of six-sided dice 100 times. The purpose is to examine the probability of how likely certain sums are rolled, and if any common patterns show. The experiment is done through the site rolladie.net, for convenience and time sake. For 100 percent accuracy, results are noted the moment they’re given.

I expect that the most common sum from this experiment will be the median of 2-12, which are all the possible sums you can roll in this experiment. The median, 7 in this case, possesses the highest number of combinations possible(6 combinations total), making it much more likely to occur during rolls. Therefore, due to the larger number of combinations resulting in a sum of 7 than any other number from 2-12, I expect a sum of 7 to be the most common result. On the other hand, numbers like 2 and 12 only possess one sum combination, leading me to believe that they’ll occur much less frequently than any other sums.

**Materials & Methods**

- Computer or mobile device
- Web browser that can access rolladie.net
- Reliable Internet Connection

- On a computer or mobile device with internet, access the website rolladie.net
- Change the amount of dice to “2″, and keep as a 6 sided dice
- Set number of rolls to “100”, and click “Go”
- Record the sums obtained from each roll of the two dice
- Analyze sums to determine least and most common outcomes, and compare results to hypothesis

**Results**

In this experiment, I explored dice roll probabilities, to find patterns and outcomes in rolling a pair of dice 100 times. After experimenting and analyzing the results, I found that there indeed are patterns that exist. The number 7, which has the most sum combinations possible, appeared as the most frequent outcome, while 2 and 12, which has the least sum combinations possible, appeared the least. Below are detailed charts and graphs of the experiment results.

Figure 1 – Table chart of first 10 rolls in experiment. For full table of 100 rolls, see link in “Appendix”(Page 6)

Figure 2 – Bar graph for results of all 100 rolls. All possible sums in experiment(2 to 12) correspond to a color(see right of graph). Sums of two dice(x-axis) start from 2 and end at 12, from left to right.

**Analysis**

After comparing the results to my hypothesis, it is clear that the outcome matched my initial expectations. As expected, the sum of 7 appeared as the most frequent outcome, at a high 23 times, while the second highest sum only went to 14. Sums close to 7(5,6,8,9) also appeared often, and were the other most common outcomes in the experiment. This observation suggests that numbers near the median, tend to also be rolled more often, due to them having more sum combinations than others. Additionally, my prediction about the sums of 2 and 12 also proved correct, as they appeared least in the experiment. The sum of 12 only appeared twice, while the sum of 2 appeared three times. These are the two lowest sums in the dataset, and can be attributed to the limited number of combinations that result in those outcomes.

Comparing my results to a previous study titled “Investigation of probability distributions using dice rolling simulation” by Stanislav Lukac & Radovan Engel from Pavol Jozef Safarik University, Slovakia provided me with more valuable information for my research. While my experiment employed only two dice and 100 trials, theirs involved 3 dice and 5000 trials(results can be found in appendix), giving them a much bigger dataset. In their study, they found 10 to be the most common sum, differing from my 7, as they had an extra dice to work with. However, if we do the calculations, we can see that 10, when using 3 dice, had the highest number of sum combinations and is also a median, just like my 7 with two dice. Even though different amounts of dice were used, our experiments were basically the same. Further proof was that in their experiment, the numbers around 10(8,9,11,12) also had the most common outcomes after the median, much like mine. These observations help underline the concept and core principle of probability. By understanding probability, predicting outcomes or likelihood of an event, like these experiments, can be much more accurate and precise.

**Conclusion**

Understanding probability is essential for decision making, and helps us understand how chance works. The experiments with dice rolls highlighted how probability affects outcomes, showing how numbers with higher amounts of sum combinations rolled more often than those with less. This result can be used by players for gambling games, since now they’d understand probability better and can adjust their strategies accordingly, giving them a higher winning chance. Additionally, by analyzing historic data and patterns, probability can also be used to predict future trends. Fields such as finance and economics, which frequently utilize probability, allows users to continually predict future changes, like stock market performances, market prices and inflation rates. Some further experiments that can be done exploring probability are using different types of dice, following the same methods used in this experiment. For example, seeing if using an 8 or 10 sided dice will differ from using a 6 sided one. Probability is a powerful tool, allowing us to predict the future, and helps us make smart decisions in a wide range of fields.

**Works Cited**

**Appendix**

Link to Complete Data Set(100 Rolls)

–https://docs.google.com/spreadsheets/d/10Hbz-eNWgfhAujNg4ZqPARvOlWqtIIREBk2NPOXuRiA/edit#gid=0

Stanislav Lukac & Radovan Engel Study Data Set