Most Probable Dice Sum:
Experimenting with Dice Probability
Abstract:
This dice probability experiment is about throwing a pair of dice and recording the result numbers. The purpose of this experiment is to roll the pair of dice at the same time and record the 2 numbers that are obtained from the roll in addition to their sum. The question being answered here is as follows: Which sum is the most probable and occurs most frequently after the pair of dice is rolled one hundred times? Therefore, the objective of this experiment is to find out which sum of the two numbers on the dice roll will end up being the greatest. In order to obtain my results, I actually rolled the pair of dice one hundred times and recorded both the 2 numbers obtained on the 2 dice and the sum they produced. I hypothesize that the most popular sum will be one of the middle numbers like 5 through 8 since very rarely does the dice roll on 1 or 6, as they are located at the extreme end of the spectrum.
Introductions
This experiment is all about probability. Probability, by definition from Merriam Webster website is, “the ratio of the number of outcomes in an exhaustive set of equally likely outcomes that produce a given event to the total number of possible outcomes”. It is basically the likeliness of an event taking place and in this case, I am trying to figure out which sum of the two numbers resulting from the dice roll will be the most probable in the one hundred times that the experiment is repeated. The dice I used had 6 sides each, which means both the dice had ⅙ of a chance to roll onto any number between 1 and 6.
Materials and Method (Procedure)
- 2 dice to roll
- Notebook and pen to record data in a table
The most important part of this experiment is to find not one, but 2 dice. Once that step is done, the rest of the experiment becomes very easy. Set up a table with the columns labeled dice #1, dice #2, and sum. In the column of dice #1 and dice #2, record the numbers that show after every roll, while in the sum column, just add the 2 numbers together and write that in. After the table is prepared, start rolling the dice. Make sure to keep track of every roll and not to mix up the numbers. Stop once you have reached the 100 roll requirement and start adding up the sums. Convert those numbers into percentages so the information is easily comprehensible. I also went ahead and created a graph to see if there were any visible trends present.
Results:
| Sum of the Pair of Dice | Percentage of the Sum Frequency |
| 2 | 2% |
| 3 | 7% |
| 4 | 10% |
| 5 | 17% |
| 6 | 12% |
| 7 | 18% |
| 8 | 15% |
| 9 | 8% |
| 10 | 5% |
| 11 | 6% |
| 12 | 0% |
Figure 1.This chart gives the percentage turnout of every sum that was recorded from the dice roll.
Figure 2. Seven seemed to be the most popular sum in the dice roll, with an 18% turnout, but the sum of 5 was very close as well, with a 17% turnout.
The sum of 12 did not appear at all throughout the experiment and the sum of 2 had the second lowest frequency percentage which was 2%. The sums from lowest occurrence to highest occurrence are as follows: 12, 2, 10, 11, 3, 9, 4, 6, 8, 5, and 7.
Analysis
The data obtained from the experiment was not very dispersed. The sums of 5, 6, 7, and 8 had very close frequencies and as one can see from the graph, these 4 sums have the tallest bars out of the entire graph. Nonetheless, the sum of 7 still had the highest percentage in the entire experiment, which was 18%. My results can be reasoned by a study done by Wolfram MathWorld, where it is proven why 7 is the most popular sum that results from dice rolls. Using the formula:
which can be rewritten as a multinomial series: |
Wolfram MathWorld was able to compute the probability of obtaining the numbers on the dice. According to the study, “The most common roll is therefore seen to be a 7, with probability 6/36 = 1/6, and the least common rolls are 2 and 12, both with probability 1/36.” In the Wolfram MAthWorld study, after completing some expansion, plugging in of different variables, and other mathematical operations, it was determined that the most common roll seen is 7 because it has the highest probability since it is in the middle of the numerical order. The formula expresses that 7 has a probability of ⅙ while 12 has a probability of 1/36, and because ⅙ is much greater than 1/36, 7 will have a much higher turnout rate than 12 and 2. The graph displays what is meant by the formula in the study and there is a trend present. The smaller numbers build up to the climax, which is 7. Therefore, my results are supported by this study.
Conclusion
The most probably sum was 7, as proved by the experiment. My hypothesis was mostly correct, however not specific, as I claimed that the sum of the middle numbers would have a greater frequency. Seven is one of those middle numbers and it had the highest probability percentage, after which 5 was the second highest. This was a probability experiment, so results for everyone may vary. My specific percentages did not match directly with the study that I made a comparison to, but the final result was similar because of the mathematical reasoning.
References APA
- Dice. (n.d.). Retrieved from http://mathworld.wolfram.com/Dice.html
- Probability. (n.d.). Retrieved from https://www.merriam-webster.com/dictionary/probability
