The Magic Number when Rolling Dice.

Ahmed Zaher

October 7th, 2024

Abstract:

This experiment investigates the probability distribution of the sum of two six-sided dice. We try to find the most common total by rolling a pair of dice several times and recording the results. To determine whether the actual evidence agrees with theoretical predictions and to evaluate the chance of each possible outcome (which can range from 2 to 12), the experiment utilizes statistical analysis. The findings show that some sums occur more frequently than others in this case the number that occurred the most was the number 7, which is consistent with the underlying probability related to the possible combinations of dice rolls. Through practical testing, this project advances knowledge of probability and statistics.

Introduction:

The objective of this experiment is to explore the probability distribution of the sums obtained from rolling two six-sided dice. The sums of a certain number of rolls (for example, 100 rolls) on a pair of dice were recorded. The total of all possible sums, from 2 (1+1) to 12 (6+6), was added together to determine how often each result would occur. The most common occurring sum was found using statistical analysis, which involved comparing the observed frequencies to the theoretical frequencies predicted by probability principles. Based on theoretical calculations, the sums of 7 should happen most often because there are the most combinations that result in this total (1+6, 2+5, 3+4, etc.). The experiment verified the theoretical predictions by showing that some sums did, in fact, appear more frequently than others. The results provide a useful application of mathematical theory in understanding chance and randomness by highlighting the fundamentals of probability and reinforcing the idea of statistical distributions.

Back in the day rolling dice was thought of as game of chance and pure luck, but it is in fact supported by scientific facts and statistical analysis, that there are sums that are going to occur more than others due to them having more combination than the other numbers. Such as stated by fellow researchers “Dice are considered the oldest gambling device, they have been around since 5000 BC. They are also used to teach probability.” (Sanih, Fatima, 2020).

This experiment is to prove that nothing is random but rather that everything in the world has a scientific and reasonable explanation that could be proven and backed by calculations.

 Methods and Materials:

If a pair of six-sided dice is rolled multiple times, then the sum of 7 will occur more frequently than any other sum, due to the higher number of combinations (six) that can produce this total compared to the other possible sums.To perform this experiment a pair of dice are rolled a 100 times and their sum is added and indexed. This is the conventional way of doing it, but this time a different approach was used. I implemented the same concepts using a MATLAB code that I wrote to simulate the rolling of a pair of dice 100 times and get the results.

numRolls = 100;

sums = zeros(1, numRolls);

for i = 1:numRolls
    die1 = randi(6); 
    die2 = randi(6); 
    sums(i) = die1 + die2; 
end

disp('Sums of each roll:');
disp(sums);

uniqueSums = unique(sums); 
countOccurrences = histc(sums, uniqueSums); 

[~, maxIndex] = max(countOccurrences);
mostRecurringSum = uniqueSums(maxIndex);
occurrences = countOccurrences(maxIndex);

fprintf('The most recurring sum is: %d (occurred %d times)\n', mostRecurringSum, occurrences);

Then I decided to use larger sample and roll the die 200 then 500 then 1000 times and observe the results. According to the laws of probability distribution, the larger the sample is the more accurate the results are going to be, following the rules of standard deviation.

Results and Observations: –

                  I observed that when running the code multiple times with a sample of 100 rolls the most occurring number keeps fluctuating, sometimes resulting in a number different than my hypothesis. As observed in other research performed on the same topic “The percentage of times eight was rolled was 17%. This percentage was calculated out of the 100 times the pair of dice was rolled. Two also being the lowest percent with only 4% out of 100 times the pair of dice was rolled. This is shown in figure 2.” (Medrano, Brandon, 2019). So increased the number of rolls gradually from 100 to 200 to 500 then finally 1000 times.  I observed that as the number or rolls gets larger the less frequent these fluctuations occur and when we get to 1000 rolls the result of the most occurring number is always the same.

Figure 1: Iteration #1 at 100 rolls

Figure 2: Iteration #2 at 100 rolls

Figure 3: Iteration #3 at 100 rolls

So, I indexed every iteration of the experiment I performed and put them all together to compare how the change in the number of rolls affected the results.

Figure 4: Iterations at different Number of Rolls

Figure 5: Fluctuation of Results With respect to number of Rolls

Analysis:

We can observe from this simulation that the most occurring number fluctuates between the numbers 5 to 9 at lower number of rolls but as the size of the sample increases the answer become more and more accurate resulting in the number 7 as the most occurring number. Which matches our initial hypothesis for what was predicted would be the result. 

This lab report focuses on determining the probability of the most frequently occurring sum when rolling a pair of dice 100 times and adding their values. MATLAB was used to simulate these dice rolls for 100, 200, 500, and 1000 trials. For each set, the sum of the dice was recorded, and the most common sum was identified. The results showed that the sums near the middle of the range (such as 7, which has the highest number of combinations) appeared most frequently, in line with theoretical expectations. As the number of rolls increased, the distribution of sums approached the expected probability distribution, where sums like 7 occur most often and sums like 2 and 12 occur least frequently.

Conclusion and Summary:

From this experiment we can deduce that the most occurring sum when rolling a pair of dice is in fact 7 as stated in our initial hypothesis, although it is not the consistent answer when rolling the dice for 100 times only, but rather fluctuates between 5 to 9 so in order to overcome this incident and eliminate the randomness a much larger sample is needed in. order to obtain a more accurate answer.

This experiment opens possibilities for more complex ones. Such as using more dice or maybe using a pair of 12-sided dice instead of the six-sided ones.

The results can be used to teach students about probability distributions, especially the concept of independent events and their combined outcomes. It also can be used in industries that rely on probabilistic models such as finance and insurance. This type of analysis can help in simulating complex random events and understanding their probable outcomes.

Reference:

• https://bmedengl2019.commons.gc.cuny.edu/lab-report/
• https://fatimasanih2.commons.gc.cuny.edu/lab-report/