# T2 Project

2021/5/30 T2 Project – Dice Rolls

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T2 Project – Dice Rolls

Site: IES College Moodle Course: Mathematics Book: T2 Project – Dice Rolls

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2021/5/30 T2 Project – Dice Rolls

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Description

Dice Roll Analysis

2021/5/30 T2 Project – Dice Rolls

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1. Introduction and Notation

2. Basic Analysis

3. Something to investigate

4. THE TASK 4.1. Investigation parameters 4.2. Number of total unique rolls 4.3. Probability of a single result 4.4. Probability distributions 4.5. Experimental results 4.6. Experimental statistics

5. Report contents and format 5.1. Report Contents from THE TASK 5.2. The Project Report 5.3. Statistical Analysis Spreadsheet

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While many current online and console games are open-ended, cinematic masterpieces, nearly all owe a large part of their fundamental systems to Dungeons and Dragons (D&D). D&D is a fantasy combat simulation played in a world of the Dungeon Master’s creation. The first official rules of D&D were published in 1974. D&D relies on dice created out of more than just cubes. The image above shows a set of gaming dice sold all over the world today.

D&D relied on the roll of a die or combinations of multiple dice to determine character statistics, damage calculations of weapons, the effectiveness of armour, how quickly health regenerates, and even their ability to lie. The interest in the game relies on the easily generated random results of dice rolls.

It became clear relatively quickly that some kind of standard notation to describe these dice rolls was needed. The agreed notation is of the form

where is the number of dice to be rolled and is the number of faces on each die. So, for example, a roll of a pair of standard six-sided dice is written as .

More examples:

Notation Explained

1d12 (or d12)

roll a single 12-sided die (when only a single die is rolled the 1 may be omitted)

3d8 roll three eight-sided dice

6d2 roll six two-sided dice (a coin toss where heads=1 and tails=2)

2d4+2d6 roll two four-sided and two six-sided dice

A X

2d6

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To continue through the rest of the project description, use the Table of Contents block on the right of the screen.

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It should be clear what the minimum and maximum totals are for a given roll of a single type of die. eg or

roll minimum maximum

Examples

For a roll that uses a combination of dice, the minimum remains that number of dice but the maximum is the sum of the maximum roll for each die.

Examples

roll minimum maximum

1d6 10d10

AdX A A × X

3d4 3 (3 × 4) = 12

5d8 5 (5 × 8) = 40

d6 + d4 2 (1 × 6) + (1 × 4) = 10

3d4 + 2d6 3 + 2 = 5 (3 × 4) + (2 × 6) = 24

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The question gamers needed to know the answer to was the one you will be investigating:

If the maximum roll is fixed, which combination of dice gives the highest average(mean) roll?

Another way to think of this question is to ask, is it better to roll a single die or many smaller ones.

For example a maximum roll of 12 can be achieved in many ways:

, , , ,

or combinations like .

Which of these gives the highest mean roll?

d12 2d6 3d4 4d3 6d2

2d5 + 1d2

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The following pages describe the project in some detail. While they describe what needs to be done, the Marking Scheme outlines what credit is given for each step.

Therefore before submitting your project files, you should reread these pages and the Marking Sheet: Project-MarkingSheet

Important information that must appear in either your Project Report or Statistical Analysis Spreadsheet are highlighted in a blue box like this one.

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While it is possible to analyse all possible combinations for all possible maximum rolls, that is well beyond the scope of the course. Therefore you will focus on a particular maximum roll and two roll combinations.

Before finalising your decision you should read the entire project documentation but make a decision now as to what your maximum roll will be given the following parameters:

the maximum roll should be between 13 and 30 the maximum number of dice you can roll is 6 each die must have between 2 and 20 faces

You can change your mind as you read the rest of the project but the Introduction of your project report must include the following:

the maximum roll the two roll combinations a prediction as to which roll will give the higher mean and why you believe it to be correct.

You may be tempted to wait until you have completed the investigation to make the prediction of the outcome but then it would no longer be a prediction but rather a reasoned fact. There are no marks awarded for the correctness of your prediction but there are marks in the conclusion of the report that refer back to it and what the investigation has taught you. Consider your actions carefully.

Complete the Project: Investigation Parameters Quiz once you have made your final decision. Your Home Room Teacher will discuss this with you if there is a potential problem with your choices.

rmax

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The project has two sections. The first deals with the theoretical probabilities of the dice roll outcomes.

For each roll combination, calculate the total number of unique rolls.

This can be done relatively easily by multiplying the number of outcomes for each die in the combination.

For example, if you were considering the combination then .

For a more complex combination like then

N =

3d4 N = 4 × 4 × 4 = 64

2d5 + 1d2 N = 5 × 5 × 2 = 50

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Successfully completing this step of the investigation will demonstrate an understanding of a number of probability concepts.

The most basic probability definition is

where is an event and is the sample space.

You have already calculated for each roll combination. You will use it to calculate the probability of a particular roll total .

For each roll combination, calculate the probability .

For example, if then .

To determine a single simply write out all the different combinations and count them up. For example, if one roll combination is the combinations with a value of are:

Die 1 Die 2 Die 3 1 4 4 2 3 4

2 4 3 3 2 4

3 3 3 3 4 2

4 1 4 4 2 3

4 3 2 4 4 1

P(A) = n(A)

n(ξ)

A ξ

n(ξ) = − 3ri rmax

P(r = )ri

= 12rmax = 9ri

n(r = )ri 3d4 = 9ri

∴ P(r = 9) = n(r = 9)

n(ξ)

= 10

64

= 5

32 = .15625

= 15.625%

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There is no single correct method for enumerating all the possibilities required here. Your report must describe what you did or what tool (eg a web site) you used to determine

.

Do the single roll outcomes calculated here support or contradict your initial prediction? How do the single roll outcomes fit with other experimental results throughout the project?

In this example, I wrote out all the possibilities in an Excel spreadsheet and then selected only those rolls where the sum was 9. More in depth instruction on how to do this can be found by watching the appropriate tutorial video.

n(r = )ri

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The final task in the probability section of the investigation is to graph the Probability Distribution for each roll combination.

You are encouraged to use one of the many web tools available to determine the probability of each possible roll as determining each of these probabilities analytically is beyond the scope of the course.

Three of the tools listed here allow for complex roll combinations like and one only gives results for groups of single dice types like . You will need to spend some time working out how your chosen web tool works and how to get the best result from it. The best advice is to read the documentation provided on the web site then experiment with the tools until you get the results required. You can always compare the results of two or more tools.

Tool Link SimpleCombinations Complex Combinations

AnyDice https://anydice.com/ ☑ ☑ Troll Dice

Roller https://topps.diku.dk/torbenm/troll.msp ☑ ☑

Snake Eyes https://snake-eyes.io/ ☑ ☑

Calc Site https://en.calc- site.com/probabilities/dice_total

☑ ☒

Discuss how the probability distributions support or contradict your initial prediction.

2d4 + 2d6 3d6

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You have calculated, using web tools and guided direction, theoretical probabilities for your roll combinations. Theory is one part of the process, what is required now are some experimental results that can be analysed using descriptive statistics.

For each roll combination gather data for at least 1000 simulated rolls. This data should be summarised as a frequency distribution table and a histogram in your Statistical Analysis Spreadsheet.

Rolls can be simulated using any tool from the previous section or using the Excel spreadsheet DiceRoller which simulates 100 dice rolls for a roll combination of up to six dice. You are not

expected to roll a physical set of dice 1000 times.

Experimental data obtained from one of the web tools will require you to import or copy/paste it into your Statistical Analysis Spreadsheet and summarise it in a frequency distribution table. One way to create this table uses spreadsheet functions like min(), max(), countif() and sum(). How you create the table is up to you. You may have to do some research to complete this part of the task.

The spreadsheet DiceRoller can be used in Excel on the web if you do not have it installed as an application on your device. Login to Microsoft365 using your studentid@iescollege.com credentials and open the spreadsheet using the Excel web app. You will have to run the simulation at least 10 times to get the required data for each roll combination. You are strongly advised to copy the data from the distribution table in the DiceRoller spreadsheet and then use “Paste Values” in your Statistical Analysis Spreadsheet

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For each roll combination calculate the following statistics mode, mean & median upper and lower quartiles interquartile range & standard deviation

using the appropriate functions in the Statistical Analysis Spreadsheet

You are encouraged to engage in some self-directed research to learn (or revise) how to calculate these statistics using spreadsheet functions.

Values displayed in the Statistical Analysis Spreadsheet should be the result of spreadsheet calculation and not a value typed into a cell.

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This section deals with what should be in your Project Report and the Statistical Analysis Spreadsheet.

The first part summarises everything already mentioned in THE TASK, while the second gives an overall snapshot of the Project Report and its contents.

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This is a summary of all the parts of the project that have appeared in the subsections of “THE TASK”

Section What is required Notes

Investigation Parameters the maximum roll the two roll combinations a prediction as to which roll will give the higher mean and why you believe it to be correct

Project Report Section: Introduction

Number of total unique rolls

For each roll combination, calculate the total number of unique rolls.

Project Report Section: Single Probabilities

Probability of a single result

For each roll combination, calculate the probability where There is no single correct method for enumerating all the possibilities required here. Your report must describe what you did or what tool (eg a web site) you used to determine

.

Project Report Section: Single Probabilities

Probability distributions The final task in the probability section of the investigation is to graph the Probability Distribution for each roll combination.

Project Report Section: Probability Distributions

Experimental results For each roll combination gather data for at least 1000 simulated rolls. This data should be summarised as a frequency distribution table and a histogram.

Project Report Section: Obtaining Experimental Data

rmax

N =

P(r = )ri = − 3ri rmax

n(r = )ri

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Section What is required Notes

Experimental statistics For each roll combination calculate the following statistics

mode, mean & median upper and lower quartiles interquartile range & standard deviation

Project Report Section: Calculated Statistics

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The Project Report should be written using the following sections and content guidelines.

Report Section What is required

Introduction Outline what the project intends to investigate. Include all information from Investigation Parameters. Of greatest significance is your prediction as to which roll combination will give the higher mean.

Single Probabilities Using the information from Number of total unique rolls & Probability of a single result. Explain how the two single probabilities are calculated. Do the single roll outcomes calculated here support or contradict your initial prediction? How do the single roll outcomes fit with other experimental results throughout the project?

Probability Distributions Explain where you obtained the graphs of the Probability Distributions for each roll combination How do they support or contradict your initial prediction.

Obtaining Experimental Data

Explain how you gathered the 1000+ experimental rolls for each roll combination Explain how you used a spreadsheet to create the frequency distribution tables. You should include a histogram for each roll combination.

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Report Section What is required

Calculated Statistics Include a table of the required statistiscs: mode, mean, median, & , IQR and standard deviation. Use the calculated statistics to support or contradict your initial prediction.

Conclusion The conclusion has three distinct parts: Make a final statement regarding your initial prediction. Does the evidence gathered support or disprove your prediction? The aim of the investigation is to answer the question: If the maximum roll is fixed, which combination of dice gives the highest average(mean) roll? Use the data obtained throughout the project to answer this question. Finally, given what you have learned, suggest another roll combination that gives an even greater mean roll than the ones you have investigated. This roll combination can fall outside the parameters of the investigation.

Q1 Q3

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This spreadsheet must contain, preferably in separate sheets for each roll combination,

a frequency distribution table a histogram formula calculations for

mode mean median

& IQR standard deviation

Q1 Q3