Name: Riyan Talukdar Lifat
Date: 10/22/2025
Title: Rolling the Dice — A Probability Experiment with 100 Rolls
Illustration: Two regular dices were rolled on a table. The numbers were added to find the total.
Abstract
In this lab, I rolled two dice 100 times to see if the results matched what math says should happen. Some totals should come up more often than others. For example, 7 should appear the most. After recording all 100 rolls, I counted how many times each total came up and compared them to the expected numbers using a chi-square test.
The results showed that my rolls were very close to what math predicted. The p value was 0.9938, which means there was no big difference. Number 7 came up the most, just like expected. This shows that even with only 100 rolls, real results can match probability very well.
Introduction
When you roll two dice, some totals happen more often than others. For example, 7 has the highest chance because there are more ways to make 7 than other numbers.
The goal of this experiment was to roll two dice 100 times, count how often each total appeared, and compare it with what probability says should happen. This helps test if the dice are fair and if math really works in real life.
Hypothesis: I think 7 will come up the most. The totals will follow the same pattern that math predicts, and there will be no big difference between real and expected results.
Materials
• Two six-sided dice (electronic dice simulator)
• Table to record results
• Computer to make graphs and do the chi square test
• Calculator or analysis software
Methods
1. Rolled two dice 100 times.
2. Added the two numbers to find the total for each roll.
3. Wrote down every total.
4. Counted how many times each total (2–12) came up.
5. Found the expected number for each total based on probability.
6. Used a chi square test to compare real and expected results.
7. Made tables and graphs to show the data clearly.
Results
When I rolled two dice 100 times, 7 came up the most, and totals near 7 (like 6 and 8) also happened more often. Totals far from 7 (like 2 and 12) came up the least. This fits what probability predicts for fair dice.
Table 1. Observed and Expected Totals for 100 Dice Rolls
| Sum | Observed | Expected | Obs − Exp | Std. Residual |
| 2 | 1 | 2.78 | −1.78 | −1.07 |
| 3 | 6 | 5.56 | 0.44 | 0.19 |
| 4 | 8 | 8.33 | −0.33 | −0.11 |
| 5 | 8 | 11.11 | −3.11 | −0.93 |
| 6 | 14 | 13.89 | 0.11 | 0.03 |
| 7 | 19 | 16.67 | 2.33 | 0.57 |
| 8 | 15 | 13.89 | 1.11 | 0.30 |
| 9 | 10 | 11.11 | −1.11 | −0.33 |
| 10 | 9 | 8.33 | 0.67 | 0.23 |
| 11 | 6 | 5.56 | 0.44 | 0.19 |
| 12 | 4 | 2.78 | 1.22 | 0.73 |
Expected values are based on theoretical probabilities for two fair dice multiplied by 100 rolls.
Chi-Square Test
- Chi-square value: 2.276
- Degrees of freedom: 10
- p-value: 0.9938
The high p-value shows there was no big difference between observed and expected results. This supports the fact that the dice behaved fairly.
Analysis
The results showed a clear pattern. The number 7 came up the most, while 2 and 12 came up the least. This matches what probability says should happen with two dice. The differences between my observed numbers and the expected ones were very small.
The chi square test also supported this. The p value (0.9938) means my results and the math predictions were almost the same. This shows that the dice were fair and the experiment worked well.
A study by Donald E. Knuth (1975) explains that when you roll dice many times, the results become even closer to the expected pattern. My 100 rolls already showed this. The peak was at 7, and the rest followed the expected curve.
This experiment proves that probability works in real life. Even with just 100 rolls, the results matched the math model. If I had rolled the dice more times, the match would probably be even closer.
My hypothesis was correct. The number 7 came up the most, and the pattern matched what math predicted. The dice acted fairly, and the chi square test confirmed that.
Conclusion
This experiment tested how real dice rolls match probability theory. After rolling two dice 100 times, the results showed that 7 appeared the most, and the numbers followed the same pattern predicted by math.
The high p value from the chi square test showed that there was no big difference between my results and the expected ones. This means the dice were fair and the experiment worked correctly.
Doing more rolls or using different dice could make the results even more accurate. This kind of experiment helps show how math and probability work in real life. It can also be used to teach fairness and randomness in games or real experiments
Reference
Knuth, D. E. (1975). The art of computer programming: Seminumerical algorithms (Vol. 2). Addison-Wesley.
