A Simple Twist of Fate:

What a Roll of the Dice Tells About Probability

Abstract

When rolling two dice it initially seems equally probable to roll any number between two and twelve. However, upon closer inspection those odds become skewed. Of all the possible sums seven is the most likely to occur. In this report I roll the dice 100 times in an effort to fully understand the likelihood of each sum’s occurrence. In doing so seven was clearly the most frequent sum, appearing 26% of the time.

Introduction

Some say that dice are a game of chance, but are they really? While it may seem that a roll of two dice could reach any sum, upon closer inspection it is clear that this is far from the truth. To understand the probability of dice rolls, I rolled a pair of dice 100 times and recorded their sum. I’m conducting this experiment to better understand probability and to understand what sum is the most likely to be rolled. In considering this problem I hypothesize that 7 will appear most frequently as the sum of two dice. I believe this because regardless of the value of one dice, 7 can be attained by one of the 6 sides of the other dice.

Materials and Methods

Materials:

• 2 six-sided die
• One hand to roll
• Apple Numbers to organize data and make graphs

Methods:

1. Roll the dice
2. Record sum of dice 
3. Tally data in a spread sheet
4. Convert spread sheet to table of dice sum and their probabilities
5. Graph data for better visualization
6. Analyze data

Results

Below is a table depicting the data from rolling two dice 100 times, organized by the sums of each roll (fig.1) as well a graph showing the same data in a more visually appealing manner. See the appendix for the full chart of each roll and the sum attained.

Number of Occurrences of Dice Sum

Sum of Dice	Total	Probability in Percentage
2	4	4%
3	7	7%
4	12	12%
5	7	7%
6	8	8%
7	26	26%
8	12	12%
9	11	11%
10	3	3%
11	5	5%
12	5	5%

Analysis

Through the experimentation clearly shown in fig. 2, seven prevailed as the most likely sum of two dice. As hypothesized, this is likely due to the fact that regardless of the value of the first dice, seven can be summed through the second dice. For example, if the value of the first dice is five, seven can be summed if the value of the second dice is two. Conversely, if the desired sum is three, the value of the first dice must either be one or two with the value second being either two or one. Seven is unique in its ability to be summed through any roll.

Roger Johnson, Associate Professor of Mathematics and Chair of the Mathematics and Computer Science Department at South Dakota School of Mines and Technology, uses the game of craps to depict the probability of rolling seven. In craps, the player must roll 7 or 11 to win the first round. Johnson writes, “There are 36 equally likely outcomes when rolling a pair of dice, and eight of them correspond to getting a total of 7 or 11” (2004). He goes on to state the probability of reaching those numbers is   simplified to    (Johnson 2004).

This is similar, though not exactly the same, as the data from fig. 1 where we see that the odds of rolling seven are 26%, while the odds of rolling 11 are 5%. These probabilities combine to 31%, about 9% higher than the 22% derived from Johnson’s probability.

Conclusion

After rolling the dice 100 times, it is apparent that seven is the sum of highest probability. In fact, based on the results of this experiment, you’ve got a 26% chance of rolling a seven. In the future this experiment could be replicated with dice that have more than 6 sides to determine if the same results occur, although the sum with highest probability would have to correlate with the number of sides of the dice. This information could also be used in playing various games to help the users win in games of chance. Regardless of how this data is used, remember that the next time you roll seven to blame it on a simple twist of fate.

Reference List

Johnson, R. W. (2006). Illustrating basic probability calculations using ‘Craps.’ Teaching Mathematics and Its Applications, 25(2), 97–103. https://doi.org/10.1093/teamat/hri012

Appendix

Full Chart of Dice Rolls

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