Abdullah Altamir 

Writing For Engineering 

Mr. Bubrow 

10/08/2024 

Exploring Probability: Analyzing the Results of Dice Rolls 

Abstract 

This report aims to explore the depth of probability and its relationships. To perform this, an experiment involving a pair of dice was performed, where we performed 100 trials of rolling a pair of dice once and calculated the sums for each trial. By analyzing the frequency of those sums, we will gain understanding whether probability is something that stays constant throughout events, or whether it fluctuates per event. From the results of our bar graph, it seems that probability is not fixed throughout each sum, but rather fluctuates as a number. 

Introduction 

Have you ever tried manipulating your luck, or more specifically, your chance of winning? What if I told you that there’s an actual science behind this question, one where it’s possible to calculate your odds at winning? This is something that will be explored in this experiment, where we will be testing the idea of dice probability using dice rolls and their sums. However, before we even begin to try to understand dice probability, we need to establish what the purpose of probability is. According to the National Institute of Health, probability “describes the possibility of an event to occur given a series of circumstances.” Probability can give us a measurement of how many times we may observe something happening in an event. In this case, for the dice probability experiment, we want to see if it’s possible that our results produce equal probabilities for each sum. Therefore, we are hypothesizing that if we roll a pair of dice 100 times, then each sum amount will have an equal probability. 

Materials and Methods 

Materials we’ll be needing for this experiment: 

• A pair of dice/ Online dice generator 

• Excel/Notebook to record observations 

The steps for this experiment: 

1. Roll a pair of dice. 

2. Record the observation on a table. 

3. Add the two numbers together to find your sum for that trial. 

4. Repeat steps 1-3 until you have performed 100 trials. 

5. Create a graph that encapsulates the sums and frequency of the sums. 

Results 

Below will be the tables and graphs that shows the results of the experiment: 

Sum of Dice Rolls	Frequency of Dice Rolls
2	2 %
3	2 %
4	9 %
5	15 %
6	15 %
7	17 %
8	17 %
9	12 %
10	4 %
11	5 %
12	2 %

Figure 1: This table gives us the percentages of every sum that was recorded. 

Figure 2: This graph shows us the spread of the calculated probability. 

Analysis 

From these results, we can see that there is not an even spread between the sums. Rather, there tends to be more of a frequency in the sums in the middle of the set {2,12}. For example, numbers 7 and 8, in the middle of the set, display some of the highest frequencies relative to the other sums. In addition to this, the sums that are the farthest/the outer points of the set, like 2 and 12, have some of the lowest frequencies. The results unfortunately debunk the hypothesis of this experiment, which was that there would be an equal probability amongst the sums, which is not the case here. It seems that probability is more nested within the middle of the set rather than being spread out amongst each sum.  

There was a similar study to this experiment where the experiment was completed using a computer program and a great number of trials. In the study, “Investigation of Probability Distributions Using Dice Rolling Simulation”, Stanislav Lukac Radovan Engel, professors from Slovakia, performed an experiment involving 5000 trials of dice rolling that were performed using three dice. The results were also like this experiment: the frequencies tended to be greater when looking at the middle of the set. When looking at the end of the sets, the frequencies were much lower, meaning the probability of rolling that sum was much lower.  

Conclusion 

Overall, the experiment gave us viable results to check our hypothesis. Unfortunately, the results disproved the overall hypothesis, which was that if we roll a pair of dice 100 times, then each sum amount will have an equal probability. The results showed us that the probabilities were higher when centered around the middle of the set, instead of being evenly spread out throughout each sum. This also coincided with the similar study, which used 3 rolls and 5000 trials to show the probability for each sum. In that experiment, there also was a higher probability for the sums in the middle of the set. Therefore, the results produced from this experiment can be applied to real life situations, such as gambling which sum of the dice rolls will pop up. Some things that could be improved in this experiment are the number of trials, which can impact the amount of sums that show up. Another thing we can do is check if rolling with a program can differ in results when rolling dice in real life.  

Reference 

1. Lukac, S., & Engel, R. (2010). Investigation of Probability Distributions using Rolling Simulations.. Australian Mathematics Teacher, 66(2), 30–35. https://web-p-ebscohost-com.ccny proxy1.libr.ccny.cuny.edu/ehost/pdfviewer/pdfviewer?vid=0&sid=e10dc3ce-c3de-4409-9334-f47e2eef8863%40redis 

2. Viti, Andrea, et al. “A Practical Overview on Probability Distributions.” Journal of Thoracic Disease, U.S. National Library of Medicine, Mar. 2015, www.ncbi.nlm.nih.gov/pmc/articles/PMC4387424/. 

Appendix 

Roll #	1st Dice	2nd Dice	Sum
1	2	5	7
2	4	2	6
3	4	3	7
4	5	4	9
5	1	4	5
6	2	2	4
7	5	3	8
8	3	1	4
9	3	1	4
10	4	5	9
11	3	4	7
12	6	4	10
13	3	2	5
14	1	5	6
15	6	1	7
16	2	3	5
17	5	3	8
18	6	3	9
19	3	4	7
20	1	3	4
21	6	4	10
22	2	4	6
23	2	1	3
24	4	2	6
25	3	5	8
26	3	5	8
27	5	3	8
28	1	1	2
29	2	1	3
30	4	4	8
31	3	5	8
32	2	4	6
33	6	3	9
34	2	3	5
35	6	6	12
36	5	4	9
37	3	4	7
38	6	5	11
39	5	1	6
40	5	1	6
41	3	2	5
42	6	6	12
43	6	4	10
44	2	5	7
45	3	6	9
46	3	3	6
47	1	5	6
48	3	1	4
49	1	1	2
50	6	2	8
51	3	2	5
52	1	5	6
53	3	4	7
54	4	4	8
55	1	6	7
56	3	5	8
57	6	5	11
58	5	6	11
59	6	3	9
60	6	1	7
61	2	3	5
62	4	4	8
63	1	4	5
64	5	6	11
65	2	6	8
66	2	2	4
67	4	2	6
68	1	3	4
69	6	5	11
70	6	1	7
71	3	5	8
72	5	4	9
73	2	3	5
74	3	2	5
75	4	5	9
76	3	1	4
77	6	3	9
78	4	2	6
79	6	1	7
80	1	4	5
81	4	1	5
82	3	2	5
83	1	5	6
84	1	5	6
85	1	4	5
86	4	4	8
87	2	3	5
88	5	1	6
89	4	5	9
90	5	4	9
91	4	3	7
92	2	6	8
93	1	6	7
94	4	3	7
95	3	5	8
96	2	2	4
97	5	5	10
98	6	1	7
99	4	4	8
100	5	2	7