Rolling a Pair of Dice for Hundred times: A study of probability of the sum

Anjali Gaba October 30, 2018

Abstract: The purpose of the experiment was to study probability of possible sum outcomes by rolling a pair of dice one hundred times.This experiment was conducted using an application on the smart phone called Simple dice Roll by ASN group LLC.It was determined that most frequent sum was 8 and also one understandable outcome was that 2 and 12 were the least frequent sums.

Introduction: to understand the probability of sum of outcomes of a pair of dice and then compare it with the theoretical probability and the experimental probability .Dice are considered one of the oldest gambling games that have attracted a lot of mathematicians to study them.1 A number of analysis of gambling are done using the concepts of probability .Probability concepts are important to almost every individual dealing with numbers.

Probability is the extent to which the occurance of an event is possible. It is measured by the ratio of the favorable outcomes to the total number of possible outcomes.

P(x)= Possible/total

For instance, an easy situation to work with would be tossing a coin, Probability of getting tails for one flip will be as:

P(tails) = {tails} = 1 = 0.5 {head, tail} 2

For the purpose of this lab report we are interested in rolling of Dice. A simple case would be dealing with one Dice and finding the probability of one particular outcome.

Probability of getting any number would be same but for numerical purposes:

P(2)= {2} = 1 ≈0.17 {1,2, 3,4, 5,6} 6

For the purposes of this lab report we need to deal with 2 dices rolling at the same time. To calculate probability of 2 events occurring at the same time we need to use :

P(A or B) = P(A) + P(B) P(A and B) = P(A)*P(B)

Since we are not dealing with and /or in this experiment these formulas aren’t that relevant but can be used in different situations.

combinations for 2 : 1,1 combinations for 3 : 2,1 combinations for 4 : 2,2; 3,1 combinations for 5 : 3,2; 4,1 combinations for 6 : 2,4; 3,3; 5,1 combinations for 7 : 3,4; 5,2; 6,1 combinations for 8 : 2,6; 3,5; 4,4 combinations for 9 : 3,6; 4,5 combination for 10 : 4,6; 5,5 combination for 11 : 5,6 combination for 12 : 6,6

Methods and Materials:

A mobile Application named Simple dice Roll by ASN group LLC was used instead of real dice as to do with real dice some space and 2 dices are required but this can be done at anyplace.the following steps were taken to perform the experiment.

1. The application was opened and 2 dice roll was selected.
2. The dice were asked to roll and the sum of outcomes was recorded in excel

   spreadsheet .

3. Step 2 was repeated for 100 times and the data was recorded.

Results:

The data for all 100 trials have been attached please refer Appendix.

Figure 1. Is a bar graph representing the frequency of each sum after 100 dice rolls. X-axis represent the the sum of the roll and y-axis represent the number of times each sum came. As evident from the graph the most frequent sum was 8

and least frequent sums were 2 and 12. As seen in the figure is roughly a bell curve withe maxima at 8 with frequency of 20.

Figure 2. Shows the pie chart representation of the data as percentages as for some people it is easy to understand using percentages rather than a bar graph.

Analysis:

As seen in the study done by Stanislav Lukac & Radovan Engel Pavol of Jozef Safarik University, Slovakia. There data results are similar to the one we got. As the sum changes to outliers and the number of possible combinations adding upto the sum decreases the probability of the sum increases as well. In the study they used 3 dice and did 5000 trials which also showed that increasing the number of trials decrease the standard deviation which means the results get more precise.

errors: in our data there are some ways that the experimental data can be more realistic by doing more number of trials. Also doing it manually or using technology to roll dice doesn’t make any difference. The uneven shape of the the bell curve is due to the uncertainties of real world.

Conclusion :

As from our hypothesis the most frequents sums lie in the range of 6-8 as they had most combinations. Our data is similar to the hypothesis except for the fact 9 has more outcomes than 6. These all are the uncertainties of real world as the theoretical probabilities are more perfect which real world is not. However , we got a roughly shaped bell curve that states that the data is trustable and roughly reproducible. As everyone rolls dice differently and could get different outcomes . But the probability of get similar results is high.

Work Cited:

1Lukac, Stanislav and Radovan Engle. 2010. Investigation of Probability Distribution Using Dice Rolling Simulation. Australian Mathematics Teachers Inc.