A Probability Analysis of Dice Sums

Ruijie Xue

Course: College Writing for Engineers and Computer Scientists

Instructor: Mr. Bubrow

Date: October 22, 2025

“A Probability Analysis of Dice Sums”

Abstract

This experiment studies the probability distribution that occurs when two uniform six-sided dice roll repeatedly. A total of 200 throws were made, and the sum of the two dice was recorded to compare the experimental and theoretical probabilities. The results show that the sum of 7 occurs at the highest frequency, while the sums of 6 and 8 occur almost at the same frequency but slightly lower than 7, which is in line with the theoretical expectation based on the number of possible combinations of each sum. Due to the limited number of trials, the minor deviations were attributed to random variations.

Introduction 

Probability is the study and measurement of the likelihood of an event occurring. It provides a method to predict uncertain outcomes and is an important tool for data analysis and decision-making. In this experiment, we study probability by rolling two dice. The purpose of the experiment is every possible and occurring probability. My hypothesis is that the frequency of the sum of 7 should be the highest, while the probabilities of the sum of 6 and 8 should be equal, both slightly lower than 7. In the following, I will describe my research methods, present the experimental results, analyze our findings, and discuss the conclusions.

Materials and Methods

Material

1. Two six-sided dice
2. A spreadsheet used for recording data
3. A calculator for calculating totals and percentages

Method

1. The method involves rolling the dice 200 times under the same conditions.
2. Record the value of the dice and the total sum after each roll.
3. Count the number of times each sum appears.
4. Calculate the experimental probability of each sum using the following method:

P= Number of occurrences​/ 200 

5. Compare the experimental results with theoretical probabilities for two dice.

Results

This section presents the overall experimental results of 200 dice rolls. We mainly focus on the probability of 7 occurring and the frequency of 6 and 8 occurring.

	sum	Frequency	Probability
	2	6	0.03
	3	12	0.06
	4	16	0.08
	5	20	0.1
	6	26	0.13
	7	34	0.17
	8	28	0.14
	9	20	0.1
	10	22	0.11
	11	9	0.045
	12	7	0.035
total		200	1

Figure 1 — Distribution of Dice Sums

Figure 2 –  Bar Chart of  Distribution of Dice Sums

Analysis

The results of this experiment support the hypothesis proposed in the introduction. Among the 200 dice rolls, the sum of 7 occurred most frequently (17%), while the sums of 6 (13%) and 8 (14%) were almost equal, but still differed by 1%. This result is basically consistent with the assumption. This model predicts that 7 is the most likely outcome because it can form the most combinations, namely (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Then, the combination of 6 and 8 is the same. 6 are (1, 5) (2, 4), (3), (4, 2), 7 (1, 6), (2, 5), (3, 4), (4, 3) (5, 2). However, the combination of 6 and 8 is less than 7, so the probability is smaller than 7. The slight difference between the experimental probability and the theoretical probability might be caused by the limited sample size, because 200 trials may not be able to completely eliminate errors.

A similar conclusion was reached by Lukac and Engel (2010) in their peer-reviewed article “Investigation of Probability Distributions Using Dice Rolling Simulation” published in The Australian Mathematics Teacher. They use computer technology to effectively simulate rolling multiple dice. As they wrote, “Galileo Galilei was interested in the sums of the scores on the dice and explained why the sum 10 occurs more frequently than the sum 9 when rolling three dice. The justification is based on the list of all possibilities of how the sums 9 and 10 can be obtained with the three dice. There are more possibilities for the sum 10 than for the sum 9 and this is the reason why the sum 10 occurs more frequently than the sum 9 in a large number of repeated random experiments. Analogously, median sums are the most probable when summing the scores on several dice, because there are more possibilities to obtain them.” This theoretical reasoning supports my experimental results. The experiment also shows that the median appears most frequently, and the median of the sum of my two dice is 7.

Conclusion

The experimental probability is consistent with the theoretical expectation when two dice are rolled. The results show that the sum of 7 occurs at the highest frequency, while the sum of 6 and 8 occurs almost at the same frequency but at a slightly lower frequency. These findings confirm the initial hypothesis and indicate that statistically, more possible combinations of results are more likely to occur.

The consistency between experimental data and theoretical models enhances the reliability of probability theory in predicting actual random events. As supported by Lukac and Engel, the median of and is the most likely because they can be obtained through the most methods and the resulting images are close to a normal distribution. Future research can increase the number of dice rolls, including computer simulations, to make the experimental results more accurate.

References

1. Lukac, S., & Engel, R. (2010). Investigation of Probability Distributions Using Dice Rolling Simulation. Australian Mathematics Teacher, 66(2), 30–35.

Appendix

Data from rolling two dices 200 times

Roll #	Die 1	Die 2	Sum	Roll #	Die 1	Die 2	Sum
1	2	5	7	100	1	3	4
2	6	6	12	101	2	1	3
3	2	4	6	102	6	6	12
4	4	3	7	103	4	6	10
5	4	6	10	104	1	1	2
6	2	1	3	105	4	4	8
7	3	1	4	106	6	6	12
8	6	1	7	107	5	6	11
9	1	4	5	108	6	3	9
10	2	1	3	109	5	2	7
11	3	5	8	110	3	5	8
12	1	6	7	111	5	3	8
13	4	6	10	112	3	5	8
14	2	2	4	113	5	2	7
15	4	5	9	114	2	1	3
16	6	3	9	115	6	2	8
17	5	3	8	116	3	1	4
18	2	1	3	117	2	5	7
19	2	3	5	118	5	2	7
20	5	3	8	119	5	5	10
21	5	3	8	120	2	6	8
22	3	3	6	121	2	4	6
23	1	5	6	122	6	4	10
24	3	4	7	123	5	5	10
25	5	2	7	124	3	5	8
26	2	6	8	125	1	1	2
27	4	4	8	126	3	2	5
28	6	6	12	127	5	1	6
29	1	2	3	128	2	3	5
30	1	5	6	129	6	2	8
31	5	5	10	130	2	3	5
32	4	6	10	131	3	4	7
33	4	5	9	132	1	1	2
34	4	2	6	133	2	4	6
35	4	3	7	134	1	6	7
36	6	1	7	135	6	2	8
37	2	5	7	136	2	3	5
38	2	2	4	137	4	6	10
39	2	5	7	138	2	4	6
40	3	6	9	139	2	5	7
41	1	1	2	140	3	5	8
42	6	3	9	141	6	2	8
43	3	4	7	142	6	4	10
44	3	3	6	143	2	1	3
45	4	5	9	144	3	2	5
46	3	1	4	145	4	5	9
47	4	5	9	146	3	5	8
48	4	1	5	147	1	6	7
49	2	6	8	148	2	6	8
50	3	3	6	149	1	6	7
51	3	6	9	150	6	5	11
52	2	2	4	151	4	1	5
53	2	2	4	152	5	6	11
54	1	5	6	153	1	1	2
55	4	3	7	154	2	4	6
56	5	5	10	155	4	6	10
57	2	3	5	156	5	1	6
58	2	2	4	157	4	4	8
59	5	5	10	158	5	3	8
60	2	4	6	159	4	1	5
61	3	3	6	160	5	5	10
62	1	6	7	161	1	1	2
63	5	6	11	162	5	1	6
64	2	5	7	163	6	6	12
65	5	5	10	164	5	1	6
66	4	3	7	165	5	1	6
67	2	1	3	166	4	1	5
68	5	6	11	167	3	3	6
69	4	1	5	168	6	1	7
70	2	2	4	169	2	3	5
71	3	4	7	170	2	1	3
72	5	5	10	171	4	1	5
73	2	3	5	172	2	6	8
74	2	1	3	173	1	3	4
75	6	4	10	174	4	4	8
76	3	4	7	175	5	4	9
77	3	2	5	176	4	3	7
78	3	4	7	177	6	4	10
79	1	2	3	178	1	3	4
80	3	1	4	179	2	4	6
81	4	5	9	180	1	4	5
82	6	6	12	181	5	5	10
83	6	1	7	182	3	5	8
84	6	3	9	183	5	4	9
85	1	4	5	184	5	6	11
86	5	3	8	185	4	5	9
87	5	1	6	186	4	1	5
88	4	4	8	187	3	3	6
89	5	2	7	188	5	4	9
90	5	5	10	189	6	5	11
91	4	5	9	190	4	6	10
92	5	2	7	191	4	5	9
93	5	4	9	192	5	6	11
94	5	2	7	193	6	6	12
95	3	1	4	194	6	3	9
96	2	1	3	195	2	2	4
97	5	6	11	196	5	5	10
98	1	5	6	197	6	1	7
99	2	2	4	198	3	5	8
100	1	3	4	199	4	2	6
101	2	1	3	200	2	4	6