36 Possibilities and 21 Different Combinations: Delving Into Dice Pair Rolling Probability
By Emelie Coello
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Abstract
I conducted an experiment where I rolled a pair of dice 100 times. I hypothesized that a majority of the number pairs will differ and that a majority of the sums will range from 5 to 9. 89% of the number pairs I rolled were and 11% of the number pairs I rolled were matching. 70% of the sums from each number pair ranged from 5 to 9 and 30% of the sums from each number pair ranged from either 2 to 4 or 10 to 12.
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Introduction
The experiment consists of rolling a pair of dice 100 times on a table. It serves as a way to observe how probability relates to dice rolling. Out of the 36 possibilities involved in rolling a pair of dice, only 6 can have a matching pair of numbers. The structure of dice makes it so that numbers that add to seven are on opposite sides. One and six, two and five, and three and four all add to seven. My hypothesis is that my pair of dice will have numbers different from each other for most of the rolls and the sum of each roll will be seven plus or minus two.
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Materials
- Two dice of equal size
- A flat surface to roll the dice on
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Methods
- I rolled two dice of equal size onto a table
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Results
I used pie charts for both Figure 1 and Figure 2 in order to display the type of number pairs I rolled and the ranges of their sums.
Figure 1: 89% of the number pairs differed. 11% of the number pairs matched.
Figure 2: 70% of the sums ranged from five to nine. Sums of 2 to 4 and 10 to 12 occurred 30% of the time.
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Analysis
The majority of the number pairs I rolled differed from each other. Figure 1 displays how 89% of the number pairs were like this while 11% were matching. The majority of the sums of the number pairs I rolled ranged from five to nine. Figure 2 displays how 70% of the sums fell in this range while 30% of them ranged from 2 to 4 and 10 to 12.
Stainslav Lukac and Radovan Engel wrote a peer-reviewed document titled, “Investigation of probability distributions using dice rolling simulation”, for volume 66 of the Australian Mathematics Teacher journal. They used three dice and found that 10 was a more frequent sum than 9 because there are more possibilities for it than there are for a sum of 9. As stated in their conclusion, “Analogously, median sums are the most probable when summing the scores on several dice, because there are more possibilities to obtain them”. In regards to my experiment, the sums with the most possibilities are 6, 7, and 8. They all have three possible combinations that add to them when two dice are being rolled together. Also, the structure of dice makes it so that numbers that add to seven are on opposite sides. One and six, two and five, and three and four all add to seven. So, I hypothesized that the number pairs I rolled will likely have sums close to seven with a margin of error of plus or minus two.
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Conclusion
Through this simple experiment, dice rolling has been connected to probability. My hypothesis proved to be correct for both of its parts. The number pairs differed more than they matched and their sums added to 5, 6, 7, 8, and 9 more than they did to 2, 3, 4, 10, 11, or 12. Multiple trials and more than two dice could be added in further experiments. The results of this experiment could be used to help people gain a basic understanding of probability.
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Appendix
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Figure 3: The number pairs I got by rolling my pair of dice 100 times
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Figure 4: Sums of the number pairs I rolled
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References
Lukac, S., & Engel, R. (2010). Investigation of probability distributions using dice rolling simulation. Australian Mathematics Teacher, 66(2), 30+. https://link-gale-com.ccny-proxy1.libr.ccny.cuny.edu/apps/doc/A229718040/AONE?u=cuny_ccny&sid=bookmark-AONE&xid=ef400cc7
