Lab Report

Does the Winning Sum of two Dice Depend on Track Length?

Colin Foster and David Martin’s hypothesis in Two-dice Horse Race is tested.

Abstract: The article Two-dice Horse Race by Colin Foster and David Martin, theoretically explains and makes a hypothesis that the probability of a horse from 2-12 winning depends on the track length where the number of the horse is the sum of the two face values on two dice and the track length is the number of tosses. However, this paper argues that the probability of a horse winning does not depend on the track length. To prove this argument, two dice were tested in two different track lengths and the results show that the winning number is 6 in both cases, not 7 and the trends were similar to each other for the two different track lengths.

Introduction: Dice are used in different kinds of games, however, the most common game using dice is surely gambling. According to the article The Probability Distribution of the Sum of Several Dice: Slot Applications, “In the US casinos, slot machines are more popular than table games, occupying at least 80% of the floor space and generating 70%-75% of the gaming revenue.” The article, Two-dice Horse Race by Colin Foster and David Martin, theoretically predicts that the chances of the 7th horse winning the race increases as the track length is increased. Bets are placed on the chances of winning of the horses. Theoretically, it is explained how the track length affects the chances of different numbered horses winning the race. However, does the track length really affect the probability of the horses of different numbers winning? In other words, does the number of throws or rolls have any effect on the probability of occurrence of the sums from 2 to 12? Knowing the answer to this question can help in making better predictions or bets.

The purpose of this experiment is to test this hypothesis that the number of throws or rolls, n, affect the probability (p) of occurrence of different numbers. If p is the probability of a number winning is p, and q is the probability of that number not winning, then

p=1-q.

In this experiment, a track length of n=300 will be used for two identical dice and the results were compared to the hypothesis of the article Two-dice Horse Race.

Methods and Materials: Materials that were used in this experiment: two dice, a flat surface (a flat floor in this case) and Microsoft Excel. The two dice were tossed on the flat ceramic floor. Then the dice were picked up and again tossed on the floor. This process was repeated 300 times and for each toss, the sum of the two face values was recorded using Microsoft Excel. The tosses were labeled with trail chronological trail numbers.

Results: As mentioned before, the face values have no particular significance for the argument of this study. What is relevant to the argument of this paper is the number of occurrences of different sums throughout the experiment.

Table: or track length of 300 the sum and corresponding occurrences.

Number of occurrences

Table: or track length of 100 the sum and corresponding occurrences.

Chart: Number of occurrences for each sum of the face values.

The bar chart clearly shows the number of occurrences of different sums compared to one another.

Chart: Probability by percentage for each sum (2-12).

The pie chart of this experimental data shows the percent probability of the number of occurrences for different sums compared to one another.

Analysis:

The bar chart makes it visible that 6 is the clear winner in this experiment. On the other hand, the hypothesis that the article Two-dice Horse Race makes shows that as the probability of number 7 winning should increase as the track length increases. The data from this experiment refutes that hypothesis and thus supports the argument made in the introduction section.

Chart: The trend between the sum and frequency of occurrence for track length of 100 and 300.

Conclusion: For the purpose of better guessing in the dice game and betting, a hypothesis was made in the beginning of this study which states that the probability of any sum winning is independent of the track length. The results of this experiment supported that argument. The guessing for those games should be made based on the basic probability rules regardless of the track lengths. According to the basic probability rules and the results of this experiment, it can be concluded that numbers 6-8 have higher chances of winning compared to the other numbers.

Appendix:

Table: Face values for the two dice and their sum in each trail.

References:

Singh, A. K., & Dalpatadu, R. J. (2011). The Probability Distribution of the Sum of Several

           Dice: Slot Applications. UNLV Gaming Research & Review Journal, 15(2), 109-118.

Foster, C., & Martin, D. (2016). Two-dice horse race. Teaching Statistics, 38(3), 98-101.

doi:10.1111/test.12108

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