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 1 | Dice 2 | Sum |
1 | 2 | 4 | 6 |
2 | 5 | 1 | 6 |
3 | 3 | 3 | 6 |
4 | 1 | 5 | 6 |
5 | 5 | 1 | 6 |
6 | 3 | 3 | 6 |
7 | 3 | 3 | 6 |
8 | 4 | 2 | 6 |
9 | 3 | 3 | 6 |
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
Rolls | Dice1 | Dice 2 | Sum |
Roll 1 | 1 | 4 | 5 |
Roll 2 | 5 | 3 | 8 |
Roll 3 | 2 | 4 | 6 |
Roll 4 | 3 | 5 | 8 |
Roll 5 | 1 | 1 | 2 |
Roll 6 | 2 | 5 | 7 |
Roll 7 | 5 | 4 | 9 |
Roll 8 | 5 | 3 | 8 |
Roll 9 | 6 | 3 | 9 |
Roll 10 | 5 | 1 | 6 |
Roll 11 | 6 | 5 | 11 |
Roll 12 | 4 | 6 | 10 |
Roll 13 | 4 | 1 | 5 |
Roll 14 | 4 | 4 | 8 |
Roll 15 | 4 | 5 | 9 |
Roll 16 | 1 | 6 | 7 |
Roll 17 | 2 | 1 | 3 |
Roll 18 | 3 | 1 | 4 |
Roll 19 | 5 | 3 | 8 |
Roll 20 | 6 | 6 | 12 |
Roll 21 | 5 | 3 | 8 |
Roll 22 | 3 | 1 | 4 |
Roll 23 | 1 | 2 | 3 |
Roll 24 | 2 | 1 | 3 |
Roll 25 | 6 | 6 | 12 |
Roll 26 | 5 | 3 | 8 |
Roll 27 | 6 | 6 | 12 |
Roll 28 | 4 | 6 | 10 |
Roll 29 | 6 | 5 | 11 |
Roll 30 | 6 | 3 | 9 |
Roll 31 | 5 | 4 | 9 |
Roll 32 | 6 | 6 | 12 |
Roll 33 | 4 | 1 | 5 |
Roll 34 | 5 | 2 | 7 |
Roll 35 | 4 | 6 | 10 |
Roll 36 | 4 | 5 | 9 |
Roll 37 | 6 | 6 | 12 |
Roll 38 | 2 | 3 | 5 |
Roll 39 | 6 | 1 | 7 |
Roll 40 | 6 | 1 | 7 |
Roll 41 | 2 | 1 | 3 |
Roll 42 | 6 | 1 | 7 |
Roll 43 | 5 | 4 | 9 |
Roll 44 | 6 | 2 | 8 |
Roll 45 | 6 | 4 | 10 |
Roll 46 | 1 | 3 | 4 |
Roll 47 | 5 | 5 | 10 |
Roll 48 | 3 | 6 | 9 |
Roll 49 | 4 | 5 | 9 |
Roll 50 | 3 | 5 | 8 |
Roll 51 | 3 | 3 | 6 |
Roll 52 | 5 | 5 | 10 |
Roll 53 | 1 | 3 | 4 |
Roll 54 | 1 | 5 | 6 |
Roll 55 | 6 | 2 | 8 |
Roll 56 | 1 | 2 | 3 |
Roll 57 | 6 | 5 | 11 |
Roll 58 | 2 | 6 | 8 |
Roll 59 | 4 | 6 | 10 |
Roll 60 | 4 | 6 | 10 |
Roll 61 | 5 | 1 | 6 |
Roll 62 | 3 | 3 | 6 |
Roll 63 | 5 | 2 | 7 |
Roll 64 | 3 | 5 | 8 |
Roll 65 | 1 | 1 | 2 |
Roll 66 | 4 | 3 | 7 |
Roll 67 | 3 | 2 | 5 |
Roll 68 | 6 | 4 | 10 |
Roll 69 | 3 | 5 | 8 |
Roll 70 | 2 | 6 | 8 |
Roll 71 | 4 | 4 | 8 |
Roll 72 | 3 | 3 | 6 |
Roll 73 | 4 | 3 | 7 |
Roll 74 | 1 | 3 | 4 |
Roll 75 | 5 | 4 | 9 |
Roll 76 | 6 | 6 | 12 |
Roll 77 | 1 | 4 | 5 |
Roll 78 | 5 | 5 | 10 |
Roll 79 | 2 | 2 | 4 |
Roll 80 | 6 | 5 | 11 |
Roll 81 | 1 | 4 | 5 |
Roll 82 | 4 | 3 | 7 |
Roll 83 | 2 | 2 | 4 |
Roll 84 | 5 | 3 | 8 |
Roll 85 | 2 | 1 | 3 |
Roll 86 | 1 | 6 | 7 |
Roll 87 | 6 | 5 | 11 |
Roll 88 | 4 | 5 | 9 |
Roll 89 | 6 | 2 | 8 |
Roll 90 | 3 | 4 | 7 |
Roll 91 | 3 | 1 | 4 |
Roll 92 | 6 | 2 | 8 |
Roll 93 | 3 | 2 | 5 |
Roll 94 | 4 | 2 | 6 |
Roll 95 | 2 | 5 | 7 |
Roll 96 | 6 | 3 | 9 |
Roll 97 | 4 | 3 | 7 |
Roll 98 | 4 | 4 | 8 |
Roll 99 | 3 | 3 | 6 |
Roll 100 | 3 | 1 | 4 |