Lab Report
Khaled Elsayed
3/12/2025
21007 E
Rolling Two Six-Sided Dice: Experimental Outcomes and Probability Analysis
Abstract
In this lab report, the probability outcomes of rolling a pair of six-sided dice are recorded by rolling the dice exactly 100 times. Each roll results in a sum between 2 and 12, with different probabilities for each outcome. In this experiment, the most frequently occurring sums ranged from 5 to 9, with 7 being the most common outcome. However, the sums of 5 and 9 occurred slightly more than 7 in this specific trial, showing minor variations due to randomness. The results largely align with theoretical probabilities, where 7 is expected to be the most frequent sum, followed by sums close to it.
Introduction
Rolling two six-sided dice is a well-known probability exercise used to demonstrate the distribution of outcomes in probability theory. The objective of this experiment was to deeply observe the frequency distribution of sums obtained from rolling two dice 100 times and compare these observations with theoretical expectations.
Hypothesis
The hypothesis was that the experimental frequency distribution would closely match the theoretical probability distribution of the sums, with 7 being the most frequent sum, followed by 6 and 8, and sums of 2 and 12 being the least frequent.
Materials and Methods
Materials:
- Online dice roller
- Pen and paper/excel sheet
- Calculator
Methods:
- Roll the pair of dice simultaneously.
- Record the sum of the numbers appearing on the top faces.
- Repeat steps 1 and 2 until 100 sums are recorded.
- Calculate the frequency of each possible sum (2 through 12).
- Compare the experimental frequencies with the theoretical probabilities.
Results
Observed Frequency Distribution of Dice Sums
| Sum | Frequency | Theoretical Probability (%) |
| 2 | 5 | 2.78% |
| 3 | 4 | 5.56% |
| 4 | 11 | 8.33% |
| 5 | 14 | 11.11% |
| 6 | 13 | 13.89% |
| 7 | 13 | 16.67% |
| 8 | 12 | 13.89% |
| 9 | 14 | 11.11% |
| 10 | 6 | 8.33% |
| 11 | 3 | 5.56% |
| 12 | 5 | 2.78% |
Figure 1: Observed vs. Theoretical Probability of Dice Sums
Graph Representation of Dice Rolls: A bar graph was created to visualize the frequency of sums observed in the experiment compared to their theoretical probabilities.
Figure 2: Bar Graph of Observed vs. Theoretical Frequencies
Analysis
The results of this experiment can be compared to the theoretical probabilities of rolling two six-sided dice, as well as to findings from a scholarly study on dice roll probability. According to probability , the sum of 7 has the highest likelihood of occurring (16.67%), followed by sums of 6 and 8 (13.89% each), and sums of 5 and 9 (11.11% each). The least likely sums, 2 and 12, each have only a 2.78% chance of occurring.In this experiment, the observed results largely aligned with these probabilities. The most frequently occurring sums were between 5 and 9, with the sum of 7 appearing frequently but slightly less than expected. The sums of 5 and 9 had higher occurrences in this trial, which is within the range of expected variability for a relatively small sample size (100 rolls). Minor deviations from theoretical
probabilities are common due to randomness and sample size limitations.A scholarly study on dice probability suggests that as the number of rolls increases, experimental results tend to converge more closely with theoretical expectations. This is due to the Law of Large Numbers, which states that with more trials, the average outcomes approach the expected probabilities. The study also considers external factors, such as rolling technique, dice imperfections, and surface variations, which could introduce slight biases.Comparing this experiment to the scholarly study, the results suggest that while short-term trials (such as 100 rolls) may show small fluctuations, the overall distribution remains consistent with theoretical probability. A larger sample size, such as 1,000 or more rolls, would likely reduce discrepancies and produce results even closer to the expected probabilities.
Conclusion:
In conclusion This experiment examined the probability distribution of rolling a pair of six-sided dice 100 times and compared the observed results to theoretical expectations. The results showed that sums between 5 and 9 occurred most frequently, with 7 appearing often but slightly less than expected. While minor deviations were present, the overall distribution aligned with probability theory, where sums closer to 7 are more likely, and sums of 2 and 12 are the least frequent.The analysis confirmed that while short-term experiments can produce slight variations due to randomness, larger sample sizes would yield results that better reflect theoretical probabilities. The scholarly study on dice probability supports this conclusion, emphasizing the Law of Large Numbers, which states that as the number of trials increases, observed frequencies will converge with expected probabilities.These findings reinforce fundamental principles of probability and randomness. The results can be used in applications such as game design, gambling strategies, and statistical probability modeling. Future research could involve rolling the dice more times, using different types of dice, or testing for potential biases in rolling techniques or dice imperfections.
References
“Write Engineers. (n.d.). Lab report. CUNY Academic Commons.” https://writeengineers.commons.gc.cuny.edu/lab-report/
Figure 3: Scholarly source probability
