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--- modding:guide:intro_dice_stats [2012/04/08 19:24] – rpc
+++ modding:guide:intro_dice_stats [2014/12/27 04:40] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
+===== Introduction to Dice Stats =====
+considering a **aDb+c** dice throw
+  * min damage: **a+c**
+  * max damage: **ab+c**
+  * average between max damage and min damage: **(ab+a+2c)/2**
+now, we want to calculate 2 arrays, the first is the array with the list of all possible throw in order: lets call this a1
+it start at **a+c** and finishes at **ab+c**
+so
+<code>
+a+c, a+c+1, ..., ab+c-1, ab+c
+</code>
+how long is this array ? **ab-a+1**
+now let's consider a second array (a2), as long as the first, and fill it with the corresponding values of the probability of the throw.
+We discovered that we can calculate the values of the probability from the diagonals of the Pascal's triangle.
+the values are exactly the values of the diagonal number a, up to the value b
+The additional values, up to the half of the array lenght or half of the array leght +1 (if the array is odd) are calculated as : number of the pascal's triangle-(number of the 1st pascal triangle at the same diagonal*a), number of the pascal's triagle-(number of 2nd pascal triangle at the same diagonal*a)
+moreover the sum of all the numbers of **a2 = b^a**
+Example, step by step:
+lets take 3d5 (aDb)
+lenght of the array is: ab-a+1=13
+a1=3,4,5,6,7,8,9,10,11,12,13,14,15
+the a (so the third) pascal's triangle diagonal is: 1,3,6,10,15,21,28,36,....
+so let's put the numbers of the diagonal up to b in the a2
+a2=1,3,6,10,15,x,y,x,15,10,6,3,1
+x will be 21-(a*1)=18
+where 1 is the first number of the same diagonal
+y will be 28-(a*3)=19
+where 3 is the second number of the same diagonal
+as a check of our calculations: 1+3+6+10+15+18+19+18+15+10+6+3+1=125
+and 125 = 5^3 (b^a)
+Resolved dices thows, as full examples:
+<code>
+d6+1
+,4,5,6,7,8,9,10,11,12,13
+,2,3,4,5,6,5,4,3,2,1
+d6+1
+,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19
+,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1
+d3
+,5,6,7,8,9,10,11,12
+,4,10,16,19,16,10,4,1
+d3
+,4,5,6,7,8,9
+,3,6,7,6,3,1
+d4
+,4,5,6,7,8,9,10,11,12
+,3,6,10,12,12,10,6,3,1
+d4
+,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20
+,5,15,35,65,101,135,155,155,135,101,65,35,15,5,1
+</code>
+Resources:
+an interactive pascal's triangle
+http://mathforum.org/dr.cgi/pascal.cgi?rows=20
+to get the diagonal
+all negative terms are 0
+the 0 term is always 1
+the nth term is the ∏ from i=1 to n of (a + i - 1)/i
+<code>
+number of the pascal's triangle-(number of the 1st pascal triangle at the same diagonal*a),
+number of the pascal's triangle-(number of 2nd pascal triangle at the same diagonal*a)
+</code>
+Just to clarify this you can have more than two extra numbers they just follow the same pattern.
+lets do some pseudo code to fill out a2 Very Happy
+lets define N(x,y) as the y term on the x diagonal
+lets define the current terms index as t
+lets define the array length as l
+<code>
+for every t up to and including l/2 - 1 + l%2
+{
+   the t term is N(a,t)-(a*N(a,t-b))
+   the l - t - 1 term is the same as the t term
+}
+</code>
+how does that look (might be an off by one error but that is the basic look)
+There's an excellent online dice roll probability calculator at http://topps.diku.dk/torbenm/troll.msp
+It has tons of features and will graph the results for you.
+===== Round off error =====
+Simple statistics unfortunately don't tell the whole story. Because Transcendence uses integer math lower damage weapons can lose a great deal to roundoff error while higher damage weapons lose proportionally less.
+Take for example the basic 1d4 laser cannon and 3d4 turbolaser cannon.
+Let's assume basic armor. Armor that has no defined resistances still has resistances and vulnerabilities derived from its level. (see [[modding/shield_and_level_damage_adjustment_curves1.01|shield and level damage adjustment curves1.01]])
+The laser resistance by level is in the hpBonus terminology used in most places in the game data
+^ Level | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+^ Armor Laser Resistance | +0 | +25 | +67 | +150 | +300 | +614 | +1150 | +2400 | +4900 | *** |
+The formula in question is 100 / (100 + hpBonus) = percentile_resistance = 1 - damage_adjustment.  Remembering that percent is literally per hundred and to get a real ratio you simply move the decimal two places more significant.
+First let's look at the laser colimnator.  It adds +75% damage.
+The laser is easy.  Just 1d4.
+^ possible rolls | 1 | 2 | 3 | 4 | average |
+^ laser | 1 | 2 | 3 | 4 | 2.5 |
+^ colimnated laser | 1 | 3 | 5 | 7 | 4 |
+As you can see it only actually gains +60%.
+The turbolaser is more troublesome with multiple dice.
+^ possible rolls | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | average |
+^ probability | 1/64 | 3/64 | 3/32 | 5/32 | 3/16 | 3/16 | 5/32 | 3/32 | 3/64 | 1/64 | - |
+^ turbolaser | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 7.5 |
+^ colimnated turbolaser | 5 | 7 | 8 | 10 | 12 | 14 | 15 | 17 | 19 | 21 | 12.75 |
+Now we're getting +70%.
+Of course doing that in a table is cumbersome.  Unfortunately you cannot algebraicly simplify a series that contains the floor function.  Machine assistance is suggested.
+Because I don't have anything like that set up right now I'm only going to do one example.  Well, two but since level 1 armor takes full rated damage from lasers that one's trivial.
+^ possible rolls | 1 | 2 | 3 | 4 | average |
+^ laser raw damage | 1 | 2 | 3 | 4 | 2.5 |
+^ against light titanium | 1 | 2 | 3 | 4 | 2.5 |
+^ against light plasteel | 0 | 1 | 1 | 2 | 1 |
+Rather than doing 60% as much damage against level 3 armor as against level 1 the laser is doing 40%, a 50% drop in effectiveness due to roundoff.
+^ possible rolls | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | average |
+^ probability | 1/64 | 3/64 | 3/32 | 5/32 | 3/16 | 3/16 | 5/32 | 3/32 | 3/64 | 1/64 | - |
+^ turbolaser raw damage | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 7.5 |
+^ against light titanium | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 7.5 |
+^ against light plasteel | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 7 | 4.09375 |
+Roundoff still hurts dropping the damage from 60% to just under 55%, but that's about a 9% reduction in effectiveness, much better than the 50% drop the laser suffers from theoretical against the same armor.
+Further examples will wait until I've made a script to do the math.