Dice Probability Experiment

Most Frequent Dice Sum: Exploring Dice Probability

Carlos Fernández
Writing for Engineering
October 7, 2024

Abstract

This experiment aimed to investigate the distribution of sums when rolling two six-sided dice
100 times. The primary goal was to determine which sums appeared most frequently and whether
the results matched theoretical probability expectations. The experiment recorded the number of
occurrences of each sum from 2 to 12. The most common result was a sum of 8, appearing 24
times. My prediction was that the sum of 7 would occur most frequently, as it has the greatest
number of possible combinations (6 out of 36 total combinations). These results provide better
understanding into the randomness and pattern of outcomes when rolling dice.

Introduction

The goal of this experiment was to explore basic ideas of probability by rolling two six-sided
dice. Each die has numbers from 1 to 6, and when rolled together, the possible sums range from
2 (the lowest) to 12 (the highest). The experiment aimed to see if the frequency of each sum
matches what we expect. Middle sums like 7 should appear most often because there are more
ways to get them, while extreme sums like 2 and 12 should appear less often because there are
fewer possible combinations to get them. The most frequent sum when rolling two dice will be 7,
as it has the most combinations. Also, the sums of 2 and 12 will appear the least often since they
have the fewest combinations.

Materials

• Data recording tool (spreadsheet)

• A random number generator (used for simulating dice rolls)

Methods
In this experiment, two six-sided dice were rolled 100 times using a random number generator.
After each roll, the sum of the two dice was recorded in a table. The frequency of each possible
sum, ranging from 2 to 12, was also represented on a chart of bars.

Results

In this experiment, the results from rolling two six-sided dice 100 times are shown in Figure 1
which lists how often each sum occurred. The data reveals that the sums between 5 and 8 were
the most common, with higher frequencies than other sums. In contrast, the sums of 2 and 12
appeared very rarely, each with a probability of about 2%, as seen in Figure 2. This pattern
shows that both the lower sums (2 to 4) and higher sums (9 to 12) have lower probabilities in this
experiment.

Figure 1. This table displays the number of occurrences for each possible sum obtained from rolling two dice 100 times.

Figure 2. This bar graph illustrates the percentage of each sum obtained from the experiment.

Analysis

Restating my hypothesis, I expected the sum of 7 to be the most frequent when rolling two dice. However, the data in Figure 1 and Figure 2 show that while 7 was common with 11 occurrences, the sum of 8 occurred most often with 24 times. The sums of 2 and 12 were the least frequent, which supports part of my hypothesis. Overall, the results suggest that sums between 5 and 8 happen more often because there are more possible ways to get them, as shown in both the table and bar graph.

Study conducted by Professor Stanislav Lukac

In Professor Stanislav Lukac’s study, he used three dice instead of two, which let him look at more possible sums. His results showed that middle sums, like 9 and 10, happened the most because there are more ways to get them with three dice. He also pointed out that extreme sums, like 3 and 18, were rare since they have fewer combinations, the same way 2 and 12 did in my own testing. In my experiment, I used only two dice, which led to different results, like the sum of 8 being the most common. Lukac’s study used thousands of rolls, giving him more accurate results compared to my smaller experiment of 100 rolls.

Conclusion

The most frequent sum in my experiment was 8, which does not match my hypothesis that 7 would be the most common. While I didn’t completely get what I expected, I correctly predicted that the sums of 2 and 12 would appear the least since they had the lowest counts. The sum of 6 was the second most common in this experiment. Overall, my results show that although 7 is a likely outcome technically being the most probable based in the number of combinations it has which is the most, 8 ended up being the most frequent. This truly highlights how random dice rolls can lead to different results each time experiments like this one are performed.

References

Lukac, S., & Engel, R. (2010). Investigation of probability distributions using dice rolling simulation. Australian Mathematics Teacher, 66(2), 30–35.
https://www.calculator.net/dice-roller.html