Hunting For 7’s: An Experiment on Dice Probability

Abstract

Probability is a concept that is applied to many areas of study, such as computer science, investment, and mathematics among others.  In this experiment, a pair of dice is rolled exactly 100 times; on each roll, the sum of the dice is recorded. Based on the rolls recorded, the sum of the dice equaled seven the most compared to other rolls, with a total of 18. However, the sums equaling eight, nine, and ten occurred frequently as well. The lab shows that the sum of the dice has a higher probability of equaling seven, as well as numbers that are close to it. 

 

Writing For Engineering 

Queen Carrasco 

8 March 2023

 

Introduction

On Khan Academy, probability is defined as how likely something is to happen. It is used in things such as forecasting the weather, traffic signaling, determining the end result of a sports game, and even when thinking of one’s chances of winning the lottery (Probability in Real Life | Applications of Probability, n.d.). 

In this experiment, two six-sided dies were used to record the probability that the dies would produce certain sums. If two six-sided dies are rolled 100 times, then there is a high probability that the sum of those dies will equal seven.

 

Materials 

• A pair of (online) dice 
• Die randomizer
• Spreadsheet to record data 
• Calculator 

 

Methods

Using the die randomizer, enter your parameters of 2 six-sided dice and 100 rolls. Select randomly. Roll for approximately 10-15 seconds before selecting a stop. Either export the spreadsheet or manually enter the data yourself. Create four columns: Roll Number, Die 1, Die 2 and Sum. For each column, enter the appropriate columns. If your spreadsheet is digital, use the provided functions in the software to calculate the sum. If not, use the calculator and record the data. 

 

Results

 

As displayed in Figure 1, the sum of 2 occurred once. The sum of 3 occurred eight times. The sum of 4 occurred ten times. The sum of 5 occurred nine times. The sum of 6 occurred thirteen times. The sum of 7 occurred eighteen times. The sum of 8 occurred eleven times. The sum of 9 occurred eleven times. The sum of 10 occurred eleven times. The sum of 11 occurred 3 times. The sum of 12 occurred five times. 

Fig 1. Occurrence of Sums of the Two Die in 100 rolls, from Sums 2-12

 

Sum of the Pair of Dice	Percentage of Sum Frequency
2	1%
3	8%
4	10%
5	9%
6	13%
7	18%
8	11%
9	11%
10	11%
11	3%
12	5%

Table 1. Frequency of Sums Out of 100 Rolls of two-sided die 

 

Analysis 

I hypothesized that if two dies were rolled 100 times, there would be a higher probability that the sum of those would equal seven. While this was proven correct, I did not take into account the numbers that were closer to seven: eight, nine, and ten.

 

Number	Combinations	Odds	Probability
2	(1,1)	1 out of 36	2.78%
3	(1,2)(2,1)	2 out of 36	5.56%
4	(1,3)(2,2)(3,1)	3 out of 36	8.33%
5	(1,4)(2,3)(3,2)(4,1)	4 out of 36	11.11%
6	(1,5)(2,4)(3,3)(4,2)(5,1)	5 out of 36	13.89%
7	(1,6)(2,5)(3,4)(4,3)(5,2)(6,1)	6 out of 36	16.67%
8	(2,6)(3,5)(4,4)(5,3)(6,2)	5 out of 36	13.89%
9	(3,6)(4,5)(5,4)(6,3)	4 out of 36	11.11%
10	(4,6)(5,5)(6,4)	3 out of 36	8.33%
11	(5,6)(6,5)	2 out of 36	5.56%
12	(6,6)	1 out of 36	2.78%

Table 2. Combinations and Probabilities of Crap Rolls (Smith & Scott, 2018)

 

Donald R. Smith, Associate Professor at the Department of Management and Decision Sciences at Monmouth University, alongside Robert Scott III, Professor at the Department of Economics, Finance and Real Estate, calculated how much control a craps shooter must possess on dice outcomes to eliminate the house advantage. Because Craps is played with someone throwing/rolling a pair of dice, the probabilities of the sums calculated would follow the same  dissemination. When comparing my results to the probability percentages from Smith and Scott (2018), shown in Table 2,  I was surprised by the slight differences in the probabilities found for sums eight, nine and ten. My results for those sums were identical, equaling 11%. 

 

Conclusion

While I was correct in regards to the high probability of the sum 7 occurring, I did not account for the discrepancy in the spread of the sums. Numbers that are close in range to 7, such as 8 and 9 can produce percentages that are similar and or identical to one another. Discrepancies in the data could possibly be attributed to the fact that I had to manually enter the data, due to issues with the software exportation system. In the future, I could use a better software system to randomize the data. Additionally, I could triple my sample pool from 100 to 300. 

 

References

1. Smith, D. R., & Scott, R. (2018). Golden Arm: A Probabilistic Study of Dice Control in Craps . UNLV Gaming Research & Review Journal, 22(1). https://digitalscholarship.unlv.edu/grrj/vol22/iss1/1
2. Probability in Real Life | Applications of Probability. (n.d.). Cuemath. https://www.cuemath.com/learn/mathematics/probability-in-real-life/
3. Probability: the basics (article) | Khan Academy. (n.d.). Khan Academy. https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/a/probability-the-basics

 

Appendix

 

Table of 100 Rolls And Their Outcomes

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