Lucky Number 7 Is “Probably” Not So Lucky: Understanding the Accuracy of Theoretical Probability Using Dice

 

Abstract

            This experiment analyzes how rolling dice can explain the concept behind theoretical probability. It was proven that when a pair of dice is rolled 100 to 200 times a probability distribution will favor a combination of numbers that have a summation of 7. There was a small percent error for the likelihood of rolling a combination of numbers with a summation of 7 (proving the accuracy of theoretical probability) and a distinct convergence of data points for a larger set of data (i.e. more dice rolls).

Introduction

            For this experiment, a pair of dice (both six-sided, numbered from 1-6) will be rolled 100-200 times and the numbers for Dice 1 and Dice 2 will be recorded. The main objective of the dice roll experiment is to observe any significant trends in data and to further analyze the results. A basic understanding of the fundamental principles of probability are also important for interpreting the results of this experiment.

            There are 36 possible combinations of numbers when rolling a pair of dice. After creating a table with all the possible combinations, the sum of these values can be calculated. It can be noted that the summation value of 7 has the highest probability when rolling Dice 1 and Dice 2 (See highlighted values in Figure 1.). The summation of 7 can occur 6 times out of the 36 possible combinations. This means that the probability of rolling a pair of dice that gives the summation of 7 will be 1/6. However, this does not mean that if a pair of dice is rolled 36 times, 6/36 times a summation of 7 would occur. Because a dice roll is random, rolling the summation of 7 6/36 times will not be the definite outcome. However, it can be hypothesized that rolling the summation of 7 with a probability close to 6/36 times is a “likely” outcome.

            In order to differentiate the similarity between rolling the combinations (1,6) vs. (6,1), (2,5) vs. (5,2), and (3,4) vs. (4,3) a red dice was used to represent Dice 1, and a blue dice was used to represent Dice 2.

Numbers Rolled	1	2	3	4	5	6
1	(1,1) S=2	(1,2) S=3	(1,3) S=4	(1,4) S=5	(1,5) S=6	(1,6) S=7
2	(2,1) S=3	(2,2) S=4	(2,3) S=5	(2,4) S=6	(2,5) S=7	(2,6) S=8
3	(3,1) S=4	(3,2) S=5	(3,3) S=6	(3,4) S=7	(3,5) S=8	(3,6) S=9
4	(4,1) S=5	(4,2) S=6	(4,3) S=7	(4,4) S=8	(4,5) S=9	(4,6) S=10
5	(5,1) S=6	(5,2) S=7	(5,3) S=8	(5,4) S=9	(5,5) S=10	(5,6) S=11
6	(6,1) S=7	(6,2) S=8	(6,3) S=9	(6,4) S=10	(6,5) S=11	(6,6) S=12

Figure 1. Dice Combinations and Sum of Values

Methods and Materials

Materials

• Red Dice
• Blue Dice
• Computer – Excel Spreadsheet

Procedure

• Create a new Excel Spreadsheet Create a column for Dice 1 (red dice) and a column for Dice 2 (blue dice).
• Roll the dice 200 times and record the number rolled for Dice 1 and Dice 2 in the respective columns.
• Create a new column and label it “SUM.” Utilize the summation function on excel to calculate the sum of each combination you rolled.

 

• Create a column and label it “Numbers.” This will represent the range of summations for all possible dice combinations. The range is from 2 to 12 (lowest combination value being (1,1), summation = 2 and highest combination value being (6,6), summation = 12)
• Create a column and label it “Frequency.” Utilize the frequency function on excel to calculate how often the summation of each value occurs.

 

 

*Note: The number range in the spreadsheet here include numbers 0 and 1, however these numbers can be omitted.

• Create a column and label it “Combination Types for the Summation of 7.” (This is based on the 6 different combinations of dice rolls from Table 1.)
• Create a column and label it “Frequency.” Count the number of times you rolled each type of combination for the summation of 7.
• Calculate the overall number of rolls for the summation of 7. Remember this value “x” will be out of 100 times.

 

 

• Compare the theoretical value of rolling a summation of 7 (theoretical probability = 1/6) to the experimental probability that was calculated in the experiment. Analyze your results.

 

Results

 

Figure 2. Frequency of Dice Roll Combinations for the Summation of 7

 

 

Figure 3. Frequency of Summation Values for 100 Rolls

 

 

Figure 4. Frequency of Summation Values for 200 Rolls

Analysis

            Figure 2. shows how often a certain combination of numbers was rolled for the summation of 7. If we consider the pairs (6,1) & (1,6), (3,4) & (4,3), and (2,5) & (5,2) to be the same thing, these combinations have about the same frequency.

            The results of the experiment prove our hypothesis correct – rolling the summation of 7 with a probability close to 6/36 times is a likely outcome. The theoretical probability 6/36 can be translated into decimal form, which is approximately 0.167.  In Figure 3., the summation of 7 occurred 17/100 times, giving a probability of 0.170. In Figure 4., the summation of 7 occurred 35/200 times, giving a probability of 0.175. The percent error for each trial, rolling 100 and 200 times can be estimated using the percent error formula.

 

            For 100 rolls, there is only a 2% percent error, and for 200 rolls there is only a 5% percent error. The percent errors exist because a dice roll is random, however, the percent errors are small because the theoretical probability still dictates the likelihood of an outcome.

            According to “Two-dice horse race,” the combination of numbers to achieve a summation of 7 is not the only thing that is important when analyzing the probability of this value. For the horses, it was very important to consider the length of the track [1]. In this experiment, the “length of the track” would signify the importance of rolling the dice enough amount of times to be able to analyze a trend in the data.

Conclusion

            Even though the theoretical probability demonstrates that the summation value of 7 is favored, you will not always get the expected results if only a few events are examined. It can be noted that the data set will converge when there is an increase in sample size. It may be small, but the data converges slightly better for 200 rolls as seen in Figure 4., as opposed to only 100 rolls as seen in Figure 3. If more experiments were conducted with more dice rolls, it would result in a more accurate representation of the theoretical probability, where the outcome will show a higher favoring for the summation of 7.

References

1. Foster, C., & Martin, D. (2016). Two-dice horse race. School of Education, University of Nottingham, Nottingham, England,98-101.

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**Lab Report Link: Sabrina McCarthy Dice Lab Report