What are the Chances of That?

The Outcomes of Rolling a Pair of Dice

 

Abstract: Throughout this experiment, I rolled a pair of dice 100 times and recorded the sum of the results for each roll. With this data, I was able to develop a conclusion that would ultimately state whether my initial hypothesis was correct or not. To conclude, I compared my own results from this experiment with another dice probability experiment to determine the similarities and differences between both sets of data.

 

Introduction: The Dice is widely regarded as one of the oldest forms of gaming dating back to 2000 B.C during the Ancient Egyptian times. The dice has played a crucial role in the development of a multitude of standard based board games all throughout history to establish a series of random numbers. The most common form of dice is cube-shaped with the numbers 1 through 6 portrayed in the form of dots on each face. Despite, the dice being so significant in the history of gaming, it has also played a key role in learning and is utilized in schools for academic purposes. In school, we use the dice in order to enable us to understand the idea of probability. Probability is the likelihood of an event occurring. In this experiment, I will be rolling two dice 100 times simultaneously and record my results for each dice. Afterwards, I will be evaluating the sum of the two dice rolled for each trial in order to calculate the probability of attaining the exact sum throughout the 100 times the two dice were rolled. In this procedure, I will be using two 6-sided dice meaning the outcome of the sum of the two dice can range between two and twelve per trial. Lastly, I will assess the number of times the same sum was rolled per trial and determine the probability of having the same sum from each roll. If two dice were rolled 100 times simultaneously, then the probability of the sum being a number higher than 6 is larger than that of the sum being a number 6 or less because the probability of a sum of 2,3,4,5 or 6 is 1/36, 2/36, 3/36, 4/36, and 5/36 respectively and that equates to 15/36, whereas the probability of the sum being larger than 6 is (36-15)/36 which is 21/36.

Materials:

• Two Dice
• Paper

Methods:

1. Roll two dice and record the numbers each dice landed on 100 times.
2. Calculate the sum of each of the two numbers for each time the dice were rolled and record it. 

Results: After conducting my experiment, I was able to collect a variety of data from each time I rolled the dice. As shown in figure 2.1, I used one red and blue dice and rolled them both 100 times and recorded my data on the table. I then added the numbers that both dice landed on to obtain the sum. From this data, I created both a bar graph and pie graph as shown in figures 2.2 and 2.3 to illustrate the frequency in which the sums of a number were between two and twelve and calculated the percent of the sum for each time I rolled the dice. Finally, I added up the number of times the sum was between one and six which was 32 and the number of times the sum was between seven and twelve which was 68. 

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

Figure 2.1- This figure depicts the results of the rolls for each dice for each trial I conducted and the sums the two numbers the dice rolled on.

 

Figure 2.2- This figure depicts a bar graph of the data I collected which shows the frequency of each sum of the dice rolled

 

Figure 2.3- This figure depicts a pie graph of the percentage of each sum of the dice rolled.

 

Analysis: Based on the data I collected, I was able to prove my hypothesis correct. My initial hypothesis stated if the dice were rolled 100 times, then the probability of the sum of both dice being between seven and twelve is greater than the probability of the sum of both dice being between one and six. Following the procedure, I took the sum of every number that landed on both dice for each trial and concluded that the sum of both dice rolled was between one and six only 32 times, while the sum of both dice was between seven and twelve 68 times. I then converted both calculations into percents and was able to determine the percent of the number being between one and six was 32%, which is less than half the amount of trials conducted. 

Ashok Singh, Rohan J. Dalpatadu, and Anthony Lucas all conducted a similar experiment in which they calculated the probability of house advantages of various bets of craps. On the contrary to my experiment in which I attempted to calculate the sums between one and six and seven and twelve, they calculated the probability distribution of the sum of k dice for (k >3). From the number rolled on each dice, they were able to derive a function of the sum x through the equation f(x)= ⅙ with x ranging between one and six. 

Conclusion: To conclude, my hypothesis was proven right and the probability of the sum being a number higher than six was greater than that of the sum six or lower. I believe this might be because of the greater mass in the dots added on each face of the dice results in the greater chance of the dice landing on a higher number. I also felt there was a sufficient amount of trials done to prove my hypothesis right and the sum of the numbers adding up to a number greater than six was a relatively high margin compared to that of a number less than six. The dice however could not always be the most reliable source of telling the likelihood of something occurring because of the various possibilities it can roll on any number. What made this experiment valid was the use of the same dice (six-sided) and the dice being rolled at the same time. I believe in the future, validity would increase if we were to use more than two dice and roll it over 100 times.

 

Work Cited: 

Singh, A.K., Dalpatadu R.J., and Lucas A.F. (n.d). The Probability Distribution of the Sum of Several Dice : Slot Applications. UNLV Gaming Research& Review Journal, 15(2), 109-118 

https://digitalscholarship.unlv.edu/cgi/viewcontent.cgi?article=1025&context=grj