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Why use dice?

• When you assume fair dice, the mechanics they are based on can be tested in hours/days rather than weeks/months

• Combinations can be used in a variety of linear and non linear ways, resulting in different gameplay experiences

• Due to our familiarity with dice in games they are often simpler to design with than just pure numbers

 

Quick terminology

• Specific types of die are called dX (d20, d4, d6)

• Dice is plural, die is singular

 

Five types of dice checks

• Rolling over

• Difficulty number is stated for a test. If the number rolled is equal to or greater than the number, the player has been successful

• Example: Checks in DnD

• Rolling under

• Same as rolling over, but player must roll equal to or under the number

• Example: Loot chests. Player must roll a d100 under their loot skill to find anything. Player’s score is 78, they rolled a 43 and passed

• Example: Sneak skill in Skyrim

• First mechanics deviation/Extended example:

Difficulty number - roll / 10 = Success level

• If a player successfully loots the chest the type of loot is dependent on the success level.

• Rolling equal

• A specific value must be rolled.

• Example: The number is 11, a player must roll a d12 and get exactly 11 to succeed. Rolling a 10 or a 12 would fail

• Rolling not equal

• The opposite of equal. Roll anything other than a specific number

• Example: The

• Values as success

• Not a test, but instead a number of successes that must be met

• Second mechanics deviation/Exploding Dice: A mechanic where the player rolls a specific value and not only gets the success, but also the chance to roll bonus dice

 

Design questions

• What should the difficulty numbers be?

• Average roll for any die : number of sides + 1 / 2

• Average of a d20 is 10 (10.5)

• You can expect a random roll of a d20 to be less than 10 50% of the time, and more that 10 50% of the time

• Difficulty numbers should be spread evenly in ascending order. (No challenge = 0, simple challenge = 5, average challenge = 10, tough challenge = 15, challenging = 20)

• What about modifiers to the rolls?

• How do modifiers improve or diminish our abilities?

• How much is too much?

• Example 1: Impress NPC by singing.

• Need more than 15. 30% Chance of success on an unaided d20 roll

• Modifier gives +2, which increases chance to 40%

• In a linear dice system each modifier represents a flat amount of probability shift. On a d20 that amount is 5% (1 in 20)

• Rolling a 14 alone is not enough, but with the modifier it is.

• In a linear system we are adding or subtracting flat rates. Designing a modifier that would change the rating of a difficulty number, we must be prepared to deal with the consequences of that for our larger games design

• What about combining dice?

• Probability of specific results is now changed. Curved probability rather than a linear one

• For example, the probability curve for a 3d6 system. Avg. value of 10 (10.5)

• Difficulty table would be different than a d20

• Probability chances of success would be far lower the further out to the curve the target number is, but can be altered with modifiers. Modifiers have more of an impact on this system than with a d20

• +2 modifier with a d20 gives a probability of 40%, while with 3d6 it is 37.46% (From respective unaided values of 30% and 16.7%

 

 

 

introduction to probability

 

Probability = The number of possible successes / the number of possible outcomes

 

What about when you want a combination?

• E.g. What is the probability of getting a 3 on a d4 and a 7 on a d8

• Probability A (3 on d4) = ¼

• Probability B (7 on d8) = ⅛

• Probability AB = ¼*⅛=1/32

 

What about when you want an option?

• Divide

 

At least

• Every probability added together that fulfils that requirement

• E.g. What are the chances of rolling at least a 4 on a d6?

• 3/6 = ½

 

Probability to not get a result

• All probabilities are numbers between 0 and 1

• Probability(not getting 3 on a 6) = 1 - (probability of getting a 3)

 

Factorial

• N-factorals

• N is a stand-in for any number

• Calculating

• Pick a positive number

• If the number is 1 or 0 the process is done

• If not, put a multiplication sign next to the number from step one

• Go back to step one and repeat the steps until you get 1

• N! = n(n-1)(...)(1)

• 5!=5 x 4 x 3 x 2 x 1=120

• 5!= 5(4!)

• N! = the number of ways we can arrange a collection of n items without repeating them

• E.g. Choosing the game of the decade

• How many ways could 3 games be arranged?

• 3 options for gold, leaves 2 options for silver, which then leaves 1 option for bronze. Which is just 3 factoral

• How many ways could 3 games out of a choice of 10 be arranged?

• Binomial coefficient = A number made by using two terms

• Bi = two

• Nomial = term

• Coefficient = a number

• Tells us how many ways without repetition there are to choose from sample of some items

• (number of items in the collection over the number of items in the sample) = the number of ways we can make choices by the sample size

• (n over r) = (n! / r!(n - r)!

• (10 over 3) = 10! / 3!(10-3)!

• (10 over 3) = 10! / 3!(7)!

• (10 over 3) = 10 x 9 x 8 / 3 x 2 x 1 (7! Cancelled out)

• (10 over 3) = 5 x 9 x 8 / 3 x 1 (10 / 2)

• (10 over 3) = 360 / 3 = 120 ways

 

 

Dice combinations - other ways of counting

 

Suppose we want to find out what the probability of rolling a 7 on a 2d6

• How many ways are there of rolling a 7?

• Enumeration (add all possible combinations by hand)

 

• How many ways can a 2d6 be rolled?

• 6 options for first, 6 options for second = 36 possible combinations

• xnxm=x(n+m)

• (x1+x2+...x6)(x1+x2+...x6) (to the power of)

• = (x2+x3+x4+x5+x6+x7) (adding all in the first equation by the first in the second equation (x1+x1, x2+x1 etc.)) (keep doing this for every expression)

• Add up all the like terms: 1 x 2+2 x 3+3 x 4…

• Coefficient tells us how many ways there are to make the roll

• 1+2+3+4… = 36

• Also tells us the probability of making that roll

• Works for different types of dice

 

 

 

Questions (answers):

 

1. Levelling could affect rolling under or over because there could be proficiencies or modifiers that make it more difficult

2. A way of imposing a punitive modifier on a linear die roll for characterisation could be having specific modifiers for certain classes/levels of player

 

1. A factoral is a stand in for a number

2. The binomial coefficient gives us a number made by two different terms

3. 11! Is 39,916,800 (11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)

4. (n over r) = (n! / r!(n - r)!

• 32 over 10 = 32! / 10!(32-10)!

• = 32! / 10!(22)!

• = 32 x 31 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 / 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

• = 4 x 31 x 3 x 29 x 14 x 9 x 26 x 5 x 4 x 23 / 9 x 7 x 4 x 1

• = 16,257,084,480 / 252

• = 64, 512, 240 possible combinations of player, very good for rotation