A Numbers Game:

Studying the Probability of a Pair of Six-Sided Dice

Abstract

By rolling a pair of fair dice 100 times, we will see how frequently the sum of the dice is equal to a number, two through twelve. What is the probability of rolling a six with a pair of six-sided dice? Through experiment and mathematical analysis we will observe the combinations and frequency of the pair of dice when the sum is equal to six. Rolling a six with a pair of dice has a higher probability of being rolled than two and twelve and has five unique ways of equaling six.

Ricardo Peralta

March 21, 2019

Introduction

Probability is the “How likely is that some event will occur or take place.” Whether you are playing a board game with your friends or gambling in a casino probability is something that is applicable in both. You can’t control probability but you can have a better understanding of what is the most likely event that will occur based on experiment and mathematical analysis. By doing so you can increase your chances of winning in a board game or winning at a casino.

In dice probability there are two factors that affect your chances of rolling any given number: the number of dice and the sides of the dice. In this experiment we will take a look at the probability of rolling a six in a pair of fair six-sided dice. When rolling a pair of six-sided dice, the probability is much more random than when rolling a one sided dice where there is a ⅙ chance of rolling any given number on a six-sided dice. When rolling a pair of six-sided the chances of you rolling a two and a twelve are very low as there are only one combination for rolling a two and twelve. In order to roll a two Dice 1 and Dice 2 have to equal 1 and in order to roll a twelve Dice 1 and Dice 2 have to equal six. The farther away a number is from two and twelve the greater the combinations of rolling that number. Therefore, since the number six is four places away from two and is six places away from twelve it should have a greater number of combinations and a greater probability of appearing rather than two and twelve if we were to do roll the dice a 100 times.

Materials and Methods

A pair of six-sided dice, with two distinctive colors, are needed for the experiment. An excel spreadsheet is preferred when recording your data.

To do the experiment label your colored dice, Dice 1 and Dice 2, where you are writing your data. Next, roll your dice 100 times; for each roll write down the outcome of Dice 1 and Dice 2 and calculate the sum of the two. Once you have your data find the frequency of outcomes by seeing how many times an outcome appeared and write it down. On excel make a histogram or pie chart to have a visual representation of the frequency of any number from two to twelve.

Results

After conducting the experiment you should see a pattern. The number of outcomes tend to grow as you move away from two and away from twelve. As seen below in Figure 1:

Figure 1

Another way to see the frequency of outcomes and the trend is by looking at a pie chart.

Figure 2

In Figure 2, as you move clockwise along the pie chart you see the pieces go from small to big and back to small.

Appearances Dice 1Dice 2Sum
1246
2516
3336
4156
5516
6336
7336
8426
9336

Figure 3

As shown in the Figure 3, we had multiple combinations from the pair of dice to get to the sum of six.

Analysis

By taking a look at our 100 rolls we can tell that as a number moves away from two and twelve the probability of that number appearing is greater and the number of combinations also increases. In addition by looking at Figures 1 and 2 you can also see the same pattern. My hypothesis was correct, since rolling a six with the pair of dice isn’t a number close to two or twelve it had a higher probability of appearing and more combinations. As shown in Figure 3, we had multiple combinations from the pair of dice to get to the sum of six, while two and twelve both only have one combination.  

While doing some more investigation I came across an article which supports my claim. An article written by Jonathan D. Baker, showed the combinations for rolling two six-sided dice for any given outcome from two to twelve. As shown below:

Rolling a six with a pair of six-sided dice has a total number of five possible combinations. Compared to my data and Baker’s combination table the number of combination actually grows as the sum of the pair of dice approaches seven from the the right of two and the left of 12.

Conclusion

As a result from the 100 roll experiment we conducted, we can say when rolling two six-sided dices that the frequency and number of combinations increases as the sum of the dice approaches seven. With this information when can determine the most probable roll to be seven when rolling a pair of six-sided dice. However, rolling dice is still random but the more times we roll the more probable it is to roll a seven because it has the greatest number of combinations. In addition, as seen from Baker’s table we can reproduce another type of table if the sides of a pair of dice increases from six-sided to seven-sided by adding a seventh column and row to his table.

References

Jonathan D. Baker. (2013). Rolling the Dice. The Mathematics Teacher, 106(7), 551-556. doi:10.5951/mathteacher.106.7.0551

Appendix

RollsDice1 Dice 2Sum
Roll 1145
Roll 2538
Roll 3246
Roll 4358
Roll 5112
Roll 6257
Roll 7549
Roll 8538
Roll 9639
Roll 10516
Roll 116511
Roll 124610
Roll 13415
Roll 14448
Roll 15459
Roll 16167
Roll 17213
Roll 18314
Roll 19538
Roll 206612
Roll 21538
Roll 22314
Roll 23123
Roll 24213
Roll 256612
Roll 26538
Roll 276612
Roll 284610
Roll 296511
Roll 30639
Roll 31549
Roll 326612
Roll 33415
Roll 34527
Roll 354610
Roll 36459
Roll 376612
Roll 38235
Roll 39617
Roll 40617
Roll 41213
Roll 42617
Roll 43549
Roll 44628
Roll 456410
Roll 46134
Roll 475510
Roll 48369
Roll 49459
Roll 50358
Roll 51336
Roll 525510
Roll 53134
Roll 54156
Roll 55628
Roll 56123
Roll 576511
Roll 58268
Roll 594610
Roll 604610
Roll 61516
Roll 62336
Roll 63527
Roll 64358
Roll 65112
Roll 66437
Roll 67325
Roll 686410
Roll 69358
Roll 70268
Roll 71448
Roll 72336
Roll 73437
Roll 74134
Roll 75549
Roll 766612
Roll 77145
Roll 785510
Roll 79224
Roll 806511
Roll 81145
Roll 82437
Roll 83224
Roll 84538
Roll 85213
Roll 86167
Roll 876511
Roll 88459
Roll 89628
Roll 90347
Roll 91314
Roll 92628
Roll 93325
Roll 94426
Roll 95257
Roll 96639
Roll 97437
Roll 98448
Roll 99336
Roll 100314