“After rolling the dice, nothing happens,” says any other regular person. As a mechanical engineering college student learning Writing for Engineering, there is a lot of science that is involved in dice rolling and I’ve learnt that information from such or any other experiment is necessary to be put into a lab report. The Twin Cube Mystery: Experiment on Dice rolling Probability ABSTRACT In this experiment, a suspected relationship between experimental and theoretical probabilities in dice rolling was investigated leading to substantial results. Using Microsoft Excel, a dice rolling simulation was performed for 200, 500 and also 1000 rolls. The frequency of the sum outcomes (from 2 to 12) was calculated as well as their experimental probability. The results were organized into tables and also represented into graphs for better analysis of the relationship between the experiment and the theory. Fair enough, the results demonstrate that as the number of rolls increases, the experimental probability closely aligns with the theoretical probability thus confirming the hypothesis. The probability pattern was also noticed in a similar experiment involving rolling three dice, suggesting a universal probability trend. INTRODUCTION “When Life and death, Victory and loss, True and False co-exist in one simple roll.” “The result of a die roll is determined by the way it is thrown, according to the laws of classical mechanics (although luck is often credited for the results of a roll). A die roll is made random by uncertainty in minor factors such as tiny movements in the thrower’s hand: they are thus a crude form of hardware random number generator.” (Dice, Wikipedia) Dice rolling provides insight to fundamental principles of probability and can be manipulated to produce even more concepts. “Dice rolls can be effectively simulated using technology. The National Council of Teachers of Mathematics (NCTM, 2000) recommends that teachers use simulations to give students experience with problem situations that are difficult to create without technology.” (Lukac & Engel, 2010) Using mathematics, it is possible to theoretically calculate the probability of landing a number or a sum of numbers in dice rolling. From this fact, this experiment was carried out to not only apply technologically simulated dice rolling but to also assess the relation between experimental dice rolling probability (physical/simulated dice rolling) and theoretical dice rolling probability (mathematically calculated dice rolling). As you roll a pair of dice, the experimental probability of the sum of the outcomes is inconsistent for the first few tries, and it doesn’t correlate to the theoretically known probability but as you drastically increase the number of rolls, the experimental(simulated) probability of landing the sums starts to resemble the theoretically(mathematically) calculated probability. MATERIALS AND METHODS The following materials were used in this experiment. ❖ Microsoft Excel Spreadsheet Procedure ● In this experiment, a pair of dice were rolled but not in the old-fashioned way. Because of the type of experiment being performed, Microsoft Excel was used to simulate rolling a pair of dice using Excel code functions. ● To roll a pair of dice, we type [=RANDBETWEEN (1,6)] as a function signifying that a random number between 1 and 6 is chosen in our value tab. This is technically similar to dice rolling because when a 6-face die is thrown, any random number between one and six can appear. ● Two columns representing two dice are made and then the number of rolls can be easily manipulated by increasing the number of rows of your two-column data. In simple terms, the number of rows corresponds to the number of rolls. ● Next, we create a column of sums and find the sum of the outcomes of the two dice by typing [=X+Y] in our value tab. X and Y represent the simultaneous outcomes of each of the dice in one roll. This can then be done for as many rows/rolls as we want. ● This was then followed by finding the frequency (number of times a number appears) of each of our possible sum outcomes (from 2 to 12) in the number of rolls we made. This frequency helped us to find the probability of landing each of our outcomes (from 2 to 12) in the rolls we made by dividing it by the total number of rolls. We then plotted a frequency to probability chart to graphically represent our experimental data and also compare it to the theoretical dice probability chart. ● We performed this procedure for the first 200 rolls, 500 rolls and then finally,1000 rolls. This was done to observe the variations in results as you increase the number of rolls. RESULTS To better understand our experiment, the following illustrations including graphs and tables were made. Figure 1. Sum, Frequency and Experimental probability of two dice rolled 200 times Figure 2. Sum, Frequency and Experimental probability of two dice rolled 500 times 4 Figure 3. Sum, Frequency and Experimental probability of two dice rolled 1000 times Figure 4. Theoretical probability column graph of the sums of a rolled pair of dice Figure 5. Experimental probability column graph of the sums of two dice rolled 200 times Figure 6. Experimental probability column graph of the sums of two dice rolled 500 times Figure 7. Experimental probability column graph of the sums of two dice rolled 1000 times ANALYSIS The two dice were rolled using Microsoft Excel for a total of 200 times in the first part of our experiment as shown in Figure 1. The corresponding Figure 5 represents the change in probability between the sums and it can be observed that the probability increases up to the sum of six then it the pattern becomes inconsistent which means that the probability is unpredictable compared to graph 1. In the second part, the number of rolls were increased to 500 rolls as seen in table 2 while calculating the probability. This brought about a noticeable change as seen in graph 3 showing us the increasing pattern up to the highest probability corresponding to that of the sum of 7 and then it started to decrease up to the last sum of 12. It can also be observed that the graph pattern starts to resemble that in theoretically derived graph1. In the last part, I increased the number of rolls to 1000 showing noticeable and expected changes. Derived from Figure 1, Figure 7 shows the same increasing pattern from the sum of 2 up to the sum of 7 and then decreasing up to the last sum of 12 as in Figure 6. But in this case a very noticeable resemblance to the theoretical in Figure 4 can be observed. Relating the three parts using the graphs in the experiment using Figures 4,5,6 and 7, it was noticed that as you increase the number of rolls while rolling a pair of dice, the experimental probability of landing the sums of the outcomes becomes more and more similar to theoretical probability pattern obtained through mathematical probability calculation. In the article, “Investigation of probability distributions using dice rolling simulation” (Lukac & Engel, 2010), while assessing the sum of the scores on the dice, Lukac and Engel find the same probability pattern as seen in graph 1 despite the fact that three dice were used in their case resulting in more probable sums. This gives proof to our hypothesis. CONCLUSION This experiment involved using Microsoft to simulate, tabulate and graph a dice rolling activity consisting of two dice. It consisted of three parts which included rolling the pair of dice a different number of times: three hundred, five hundred and one thousand times. In each of these cases, the frequency of each sum was calculated to find the probability of occurrence of each sum. These experimental probabilities were all graphed in comparison to the theoretical probability graph. After comparison, our results turned out to match our hypothesis suggesting that as we increased the number of rolls, the experimental probability of the sums became closer and similar to the theoretical probability. This theoretical probability was also compared to that of another similar study which involved rolling three dice. Not only did they exhibit the same pattern but also sparked an idea of a new hypothesis suggesting that despite the number of dice used, the same graphical probability will always be noticed when the dice are rolled a large number of times between the experimental and the theoretical. References Lukac, S., & Engel, R. (2010). Investigation of probability distributions using dice rolling simulation. Australian Mathematics Teacher, 66(2), 30–35 Wikimedia Foundation. (2024, October 1). Dice. Wikipedia. https://en.wikipedia.org/wiki/Dice