Author Topic: Optimization by the numbers.  (Read 16390 times)

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lans

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Re: Optimization by the numbers.
« Reply #20 on: March 01, 2011, 11:41:42 AM »
I take no credit for any of this. Everything below is the writing of Tleilaxu_Ghola, not mine. I did not contribute to this in any way, shape, or form barring this copy paste. And some minor formatting changes to adjust it to these boards. Either way, nothing special.

Caveats: The results of this analysis may not fit your personal idea of optimization.  I fully realize that many builds exhibit unquantifiable benefits which do not fit in the analysis I provide below.  That said, this strictly numeric analysis contains some interesting results.
Introduction:
The entire premise of this thread will be to analyze characters based on two things: the probability that a monster will succeed in resisting an attack or the probability that a character will succeed in resisting a monster's attack.  From here it should be obvious that one desires the former probability to be as low as possible and the latter to be as high as possible.  This thread will not assess any character traits that cannot be easily described by a probability.  Essentially this means I'm limiting myself to opposed rolls and DC checks.  I realize this is not a comprehensive analysis, and I will never claim that it is.  Nevertheless, it should be a good analysis for the area it attempts to assess.

The Data:
This thread concept was previously impossible and the realm of speculation because of a lack of proper data.  This has changed, two individuals (CubeKnight & Dielzen) have performed an excellent job of data mining to bring us what follows:

Average and Max Values per CR:

[spoiler]
Code: [Select]
[b] hp init AC touch ff_ac bab fort ref will[/b]
Summary for 'challenge_rating' =  1/10 (2 detail records)
Avg 1.00 1.50 15.50 15.50 14.00 0.00 2.00 3.50 2.00
Max 1 2 16 16 14 0 2 4 2
Summary for 'challenge_rating' =  1/8 (2 detail records)
Avg 1.00 2.00 14.00 14.00 12.00 0.00 2.00 3.00 0.50
Max 1 2 14 14 12 0 2 4 1
Summary for 'challenge_rating' =  1/6 (4 detail records)
Avg 4.50 1.75 13.75 13.25 12.00 0.25 2.50 4.00 1.00
Max 11 2 14 14 12 1 4 4 2
Summary for 'challenge_rating' =  1/4 (8 detail records)
Avg 3.88 1.75 14.50 13.25 12.75 0.25 2.50 2.88 0.38
Max 11 3 17 15 14 1 4 5 2
Summary for 'challenge_rating' =  1/3 (7 detail records)
Avg 4.43 2.57 15.71 13.14 13.71 0.14 2.29 3.14 1.14
Max 6 5 17 15 16 1 4 5 3
Summary for 'challenge_rating' =  1/2 (29 detail records)
Avg 6.59 1.41 14.69 11.86 13.38 0.76 2.62 2.00 0.03
Max 16 4 18 16 18 2 5 6 3
Summary for 'challenge_rating' =  1 (46 detail records)
Avg 12.24 1.54 15.28 11.78 13.80 1.28 3.00 2.85 0.98
Max 31 7 23 20 18 3 6 6 5
Summary for 'challenge_rating' =  2 (51 detail records)
Avg 20.55 2.39 15.76 11.84 13.94 2.33 4.35 4.35 2.69
Max 42 8 23 15 22 5 8 10 6
Summary for 'challenge_rating' =  3 (72 detail records)
Avg 27.26 2.68 16.14 11.51 14.64 3.30 4.58 4.18 3.31
Max 55 9 23 18 23 6 9 9 7
Summary for 'challenge_rating' =  4 (40 detail records)
Avg 48.23 2.23 16.00 10.45 15.05 5.20 6.45 5.10 4.60
Max 94 6 20 18 20 9 11 9 9
Summary for 'challenge_rating' =  5 (51 detail records)
Avg 56.33 3.16 17.16 10.55 15.73 6.08 7.47 5.82 4.82
Max 95 11 25 16 25 11 12 13 10
Summary for 'challenge_rating' =  6 (26 detail records)
Avg 69.12 3.27 18.88 11.00 17.27 7.60 8.08 6.85 6.08
Max 133 13 29 29 24 12 14 12 12
Summary for 'challenge_rating' =  7 (45 detail records)
Avg 86.98 2.91 18.07 10.38 16.44 8.38 8.67 6.98 6.56
Max 152 13 25 17 24 13 15 19 12
Summary for 'challenge_rating' =  8 (31 detail records)
Avg 96.48 3.19 20.00 10.58 18.32 9.43 9.19 7.65 7.45
Max 180 10 27 20 26 15 16 14 11
Summary for 'challenge_rating' =  9 (31 detail records)
Avg 130.65 3.61 21.74 10.45 19.68 11.97 12.13 9.81 8.58
Max 230 14 29 18 28 18 19 22 13
Summary for 'challenge_rating' =  10 (19 detail records)
Avg 136.53 2.79 22.58 9.26 21.42 13.21 11.63 8.58 9.11
Max 305 8 33 13 33 24 22 19 14
Summary for 'challenge_rating' =  11 (24 detail records)
Avg 163.83 3.96 23.71 10.92 21.38 14.25 13.75 10.63 10.3
Max 228 15 29 25 27 24 20 25 14
Summary for 'challenge_rating' =  12 (12 detail records)
Avg 196.33 1.58 21.75 7.17 21.42 17.00 15.42 9.33 9.08
Max 300 5 28 13 28 30 25 12 15
Summary for 'challenge_rating' =  13 (12 detail records)
Avg 167.00 2.92 27.33 10.25 26.00 14.42 14.17 9.83 12.4
Max 230 5 32 14 31 20 18 14 16
Summary for 'challenge_rating' =  14 (12 detail records)
Avg 180.33 4.50 27.00 11.00 25.17 16.17 15.92 11.83 14.0
Max 287 8 35 14 35 23 19 13 18
Summary for 'challenge_rating' =  15 (8 detail records)
Avg 224.63 3.50 29.75 8.13 29.50 18.88 15.75 10.63 15.8
Max 312 5 34 11 33 24 19 14 20
Summary for 'challenge_rating' =  16 (11 detail records)
Avg 239.09 4.27 31.91 10.00 30.73 21.18 17.09 13.27 16.6
Max 378 8 42 16 42 31 23 16 23
Summary for 'challenge_rating' =  17 (7 detail records)
Avg 244.86 3.29 28.00 9.57 26.86 20.14 19.57 11.29 16.2
Max 337 7 34 13 34 27 25 15 19
Summary for 'challenge_rating' =  18 (8 detail records)
Avg 302.25 6.00 32.75 8.50 31.25 24.25 19.50 15.63 19.0
Max 375 20 37 14 37 30 23 24 23
Summary for 'challenge_rating' =  19 (10 detail records)
Avg 355.80 4.80 36.00 9.20 35.20 27.60 22.10 16.30 19.7
Max 445 12 38 16 38 33 25 20 21
Summary for 'challenge_rating' =  20 (9 detail records)
Avg 409.33 6.00 36.44 9.11 34.44 29.56 24.22 18.67 21.2
Max 858 12 40 17 39 48 38 29 23
Summary for 'challenge_rating' =  21 (13 detail records)
Avg 393.00 4.31 38.69 8.77 37.31 27.69 23.31 16.15 20.5
Max 522 7 51 17 44 36 28 22 25
Summary for 'challenge_rating' =  22 (9 detail records)
Avg 452.33 7.44 40.22 11.56 36.33 29.56 23.00 19.67 23.7
Max 536 22 47 24 47 37 28 25 27
Summary for 'challenge_rating' =  23 (11 detail records)
Avg 480.09 8.27 39.91 12.45 35.27 31.00 26.00 21.82 24.6
Max 893 18 46 28 42 38 41 35 28
Summary for 'challenge_rating' =  24 (8 detail records)
Avg 622.25 9.25 42.38 12.88 37.75 37.50 29.50 24.25 27.3
Max 900 27 44 40 44 48 39 38 38
Summary for 'challenge_rating' =  25 (11 detail records)
Avg 600.27 8.55 44.18 18.36 39.82 34.09 27.55 24.00 28.7
Max 1105 26 52 50 50 40 45 36 41
Summary for 'challenge_rating' =  26 (7 detail records)
Avg 420.29 10.43 44.29 27.14 39.57 27.00 21.14 20.14 26.2
Max 680 17 50 48 43 40 32 29 32
Summary for 'challenge_rating' =  27 (4 detail records)
Avg 625.25 10.00 46.75 19.25 40.25 43.50 31.75 27.75 32.7
Max 817 20 52 32 45 75 40 37 41
Summary for 'challenge_rating' =  28 (4 detail records)
Avg 894.25 16.25 53.50 26.50 43.25 42.50 37.00 37.00 31.5
Max 1102 25 57 47 57 60 45 44 35
Summary for 'challenge_rating' =  29 (2 detail records)
Avg 472.00 12.00 50.50 30.00 42.50 21.50 25.00 25.50 24.5
Max 814 17 51 38 48 33 40 39 32
Summary for 'challenge_rating' =  30 (4 detail records)
Avg 1064.25 2.00 40.75 13.50 40.25 48.25 36.00 26.25 38.7
Max 1785 6 58 26 58 70 47 37 50
Summary for 'challenge_rating' =  31 (2 detail records)
Avg 788.50 3.50 58.00 16.00 56.50 42.50 34.00 24.50 33.5
Max 877 4 64 20 64 45 37 25 38
Summary for 'challenge_rating' =  32 (1 detail record)
Avg 433.00 18.00 40.00 20.00 30.00 42.00 19.00 29.00 29.0
Max 433 18 40 20 30 42 19 29 29
Summary for 'challenge_rating' =  33 (2 detail records)
Avg 605.50 -2.00 60.00 0.00 60.00 72.00 32.00 30.00 33.0
Max 608 -2 60 0 60 72 32 30 33
Summary for 'challenge_rating' =  34 (2 detail records)
Avg 1100.00 9.50 54.00 16.00 50.50 52.50 42.00 32.50 39.5
Max 1362 15 61 19 61 62 48 42 43
Summary for 'challenge_rating' =  35 (5 detail records)
Avg 1011.80 12.60 62.00 13.80 53.20 48.40 41.60 34.20 31.0
Max 1075 24 67 18 67 50 46 52 43
Summary for 'challenge_rating' =  36 (1 detail record)
Avg 1676.00 11.00 58.00 3.00 57.00 72.00 55.00 47.00 39.0
Max 1676 11 58 3 57 72 55 47 39
Summary for 'challenge_rating' =  37 (1 detail record)
Avg 1292.00 4.00 74.00 20.00 74.00 55.00 46.00 29.00 47.0
Max 1292 4 74 20 74 55 46 29 47
Summary for 'challenge_rating' =  39 (3 detail records)
Avg 1430.00 7.00 66.67 18.00 62.33 68.00 55.67 37.67 45.0
Max 1728 13 81 22 81 96 75 55 52
Summary for 'challenge_rating' =  41 (1 detail record)
Avg 1856.00 14.00 58.00 16.00 44.00 96.00 76.00 56.00 42.0
Max 1856 14 58 16 44 96 76 56 42
Summary for 'challenge_rating' =  42 (1 detail record)
Avg 1984.00 12.00 60.00 14.00 48.00 96.00 77.00 54.00 42.0
Max 1984 12 60 14 48 96 77 54 42
Summary for 'challenge_rating' =  43 (2 detail records)
Avg 1516.00 4.00 79.50 21.50 79.50 59.00 50.00 31.00 50.5
Max 1787 4 88 24 88 65 55 34 56
Summary for 'challenge_rating' =  48 (1 detail record)
Avg 1479.00 4.00 78.00 21.00 78.00 58.00 50.00 31.00 50.0
Max 1479 4 78 21 78 58 50 31 50
Summary for 'challenge_rating' =  50 (2 detail records)
Avg 2472.50 7.00 83.50 19.00 78.50 83.00 72.00 44.50 51.5
Max 2880 10 95 26 95 96 84 52 61
Summary for 'challenge_rating' =  52 (1 detail record)
Avg 1732.00 4.00 85.00 23.00 85.00 63.00 54.00 33.00 54.0
Max 1732 4 85 23 85 63 54 33 54
Summary for 'challenge_rating' =  57 (2 detail records)
Avg 1517.00 7.00 81.00 26.50 81.00 60.00 49.00 33.00 43.0
Max 2006 10 92 28 92 68 59 36 59
Summary for 'challenge_rating' =  59 (1 detail record)
Avg 2362.00 4.00 102.00 28.00 102.00 75.00 64.00 39.00 65.0
Max 2362 4 102 28 102 75 64 39 65
Summary for 'challenge_rating' =  61 (1 detail record)
Avg 2299.00 4.00 99.00 27.00 99.00 73.00 63.00 38.00 63.0
Max 2299 4 99 27 99 73 63 38 63
Summary for 'challenge_rating' =  66 (1 detail record)
Avg 2613.00 4.00 106.00 29.00 106.00 78.00 68.00 41.00 68.0
Max 2613 4 106 29 106 78 68 41 68
    [/spoiler]
    The above data are extracted from all monsters in the SRD; this work was done by Dielzen.

    The next data is similar, but contains numerous "CO sources" such as MMI through MMIV, stormwrack, sandstorm, and others:
>>LINK<<

The above link has a great deal more functionality than Dielzen's table, so I strongly recommend using it.  The tool is simply wonderful.

Data Analysis: There's a lot of data here and making sense of it is perhaps more of a chore than collecting it.  The first thing I will do is establish a rubric for character analysis that will make use of this data.  Once that is done, analysis of any particular character will involve nothing more than looking up values in a table and clicking away at your calculator.  The next thing I will do is go through the data and extract whatever interesting and salient pieces of information that I can.

Character Rubric:
  • Step 1 (Set up the problem): Figure out what level you would like to analyze the character.  This will determine the CR band that your character might be dealing with.  I will use the following notation for single opponents:
    • Worthless: CR = Level -5 to Level -4
    • Trash: CR = Level -3 to Level -2
    • Average: CR = Level -1 to Level +1
    • Boss: CR = Level +2 to Level +3
    • Impossible: CR = Level +4 to Level +5
    One doesn't always fight single monsters, luckily there's a good resource for assessing groups: DMG Pg 49.  Simply look up your level and you can see how many trash mobs to expect per encounter.  For example, a 20th level character can expect up to 4 16 CR creatures in one encounter and up to 12 CR 13s.  Dealing with groups is a little trickier as far as assessment goes, I'll discuss this further in step 6.
  • Step 2 (Offensive Capabilities): Find the salient attack forms of the character.  Is it a spell slinger?  A crowd controller?  A melee type?  An archer?  Whatever it does, it's probably dependent on some sort of opposed check to attack, be it saving throws or AC.  The next few sub-steps will walk you through the probability calculations.  It's really not that hard, once you get a hang of it.  Trust me. First, determine the mechanic of the offensive capability.  Is it an opposed roll (like trips, skill checks, etc), or is it a DC check (like spell saves)?  There are two fundamentally different things, the former being the most complex.
    [list=1]
  • Opposed Rolls:
    [SBLOCK=Explaination/Derviation]Opposed rolls all work the same way.  Each side rolls a d20 and adds a modifier to it.  Whoever has the highest modifier wins the tie.  Here's how I think about it (may not be the best way of solving it, but it works):
    Treat the monster's roll as a probability and yours as a fixed variable.  Examine it case by case.  Each of your rolls has a probability of 1/20, calculate the probability that the monster could beat your roll with the modifiers in place.  Multiply that probability by 1/20 and sum over all twenty events.  The total result cannot be greater than 100%.  To put it in generalized terms:
    N-1?Pr(X+n<Z+Y), where X is your modifier to the opposed roll, the sum index is n (ranging from 1 to N), Z is the monster's modifier, and Y an RV representing his roll (uniform discrete from 1 to N).

    Basically the only chore is to figure out how many possible rolls there are such that the monster succeeds, ie where X+n < Z + Y is satisfied.  We restrict outselves to the case where X-Z <N.  If Z-X >= N+1, then we always lose.  Take a concrete example, Z = +15, X = +10, and you roll a 6.  The monster needs to roll a 2 or better, leaving 19 possible rolls in which he could succeed.  For any roll of 5 or lower on your part, the monster always succeeds, contributing a total of 25% to his total probability of success.  In variables, that's (N+[Z-(X+n)]).  Plug in N = 20, Z = 15, X = 10, and n = 6, and you'll see it gives 19, that is 19 possible rolls to succeed.  Going into this we knew that we needed to roll better than (Z-X), since Z > X.  Now we have a general formula for Z > X.
    N-2?(N+[Z-(X+n)]) + N-1(Z-X), where n ranges from (Z-X+1) to N.
    Now lets evaluate the sum... this will require a few tricks.  If you don't follow... I'm sorry.  I'll write out each step.
    N-2?(N+[Z-(X+n)]) = N-2?(N+Z-X)-N-2?(n), both sums over the original range.
    N-2(N+Z-X)(N-(Z-X))-N-2N(N+1)/2 + N-2(Z-X)(Z-X+1)/2
    Bringing the total to:
    N-2(N+Z-X)(N-(Z-X))-N-2N(N+1)/2 + N-2(Z-X)(Z-X+1)/2+N-1(Z-X)

    Just to check, lets try the original case, plugging in N = 20, X = 10, Z = 15.
    (1/400)(25)(15)-210/400+15/400+100/400= 280/400 = 70%.  Meaning you would only have a 30% chance of beating a monster in this opposed check.

    Now lets look at the case where Z<X.  We will restrict ourselves to the case X-Z <N, because if X-Z >= N, we'd always win.  Again, let’s use a concrete example.  Z = 10, X = 15, and you roll a 6.  The monster has to roll a 12 to beat you, giving him 9 possible rolls to succeed.  So N-([X+n]-Z) gives the number of possible rolls.  The monster always fails for rolls less than 7, so the best he can get is a 70% success rate (he automatically fails 6/20 times).  In variables, that's (X+1)-Z automatic failures.  We, as the player, needn't roll any higher than N-(X-Z) to ensure success on our part.  I translate this to mean that the monster automatically fails for n > N-(X-Z).  We're now in position to calculate the general formula for X > Z.
    N-2?(N-[(X+n)-Z]), where n ranges from 1 to N-(X-Z)-1.
    Evaluation time... again standard manipulations to evaluate the sum
    N-2?(N-[(X+n)-Z]) = N-2?(N-X+Z]) - N-2?(n), both sums over the original range.
    N-2(N-(X-Z)-1)(N-X+Z) - N-2(N-(X-Z))(N-(X-Z)-1)/2
    Bringing the total to:
    (1/2)N-2(N-(X-Z)-1)(N-X+Z)

    Again, lets check with the variables from the original scenario to make sure I didn't screw up badly:
    1-(1/2)(1/400)(14)(15) = 1-105/400 = .7375.

    Note the above derivation assumes that the offender wins the tie, this is actually wrong in retrospect, so rather than redoing all the sums, I'm just going to calculate the tie probability.  Then we'll subtract it from the "offender".

    Same deal as before, we need to find all the possible rolls such that the condition X + n = Z + Y.  In other words X-Z = Y-n.  Obviously if Z-X > N, then the two cannot ever be equal.  Likewise, if X-Z > N, then the two cannot ever be equal.  These form the bounds for our problem.  The interesting thing is that for all the possible values of X + n (ranging from X + 1 to X + N), there is only one combination Z + Y that will satisfy the tie condition.  The only difficulty then becomes discovering the range for n.  Basically we're going to have two scenarios: your mod is bigger or the monster's mod is bigger.  If your mod is bigger then the range of n in the sum is from 1 to N - (X-Z).  If your mod is smaller, the sum ranges from Z-X+1 to N.  So basically we come up with these to formulas:
    If your mod is bigger: Tie probability = N-2*(N-X+Z)
    If your mod is smaller: Tie probability = N-2*(N-Z+X)

    Wanna check?  Take my mod to be 15 and the monster's to be 10.  Check that link (below) and you'll see that the tie probability is 3.8%.  My formula gives: (20-15+10)/400 = 15/400 = 3.75 %.  The other one is the exact same thing, just with the modifiers reversed in role.  So now our final formulas come to:

    If your mod is bigger:
    Probability of Success = 1 - (1/2)N-2(N-(X-Z)-1)(N-X+Z), where X = your modifier, Z = monster modifier

    If your mod is smaller:
    1 - (1/2)N-2[((N-(Z-X))(N-(Z-X)+1) - 2(N-Z+X)], where X = your modifier, Z = monster modifier

    In a moment of insecurity, I decided to check these numbers against what other people have said.  I googled a bit and came to this site: >>LINK<<
    His numbers agree with mine (rounded to one decimal place).  Just add the tie probability to the win probability to see.  To check the case Z > X, add the tie probability to the lose and you get the 70/30 split I calculated.
    [/SBLOCK]
    Formulas (for d20 system... N = 20).
    If your mod is bigger:
    Probability of Success = 1 - (1/2)N-2(N-(X-Z)-1)(N-X+Z), where X = your modifier, Z = monster modifier

    If your mod is smaller:
    1 - (1/2)N-2[((N-(Z-X))(N-(Z-X)+1) - 2(N-Z+X)], where X = your modifier, Z = monster modifier

    Note that if X-Z >= N, we always win.  If Z-X >= N+1 we always lose.  So these form the bounds for X and Z with respect to N.
  • Roll versus DC:
    [SBLOCK=Explaination/Derivation]
    Whew, this guy is waaaay easier to do.  For any roll versus a fixed DC where X = your modifier, Y is the random variable, and Z = the DC, we have the following situations:
    Pr(Y+X>=Z) = 1-Pr(Y+X<Z) = 1-Pr(Y<Z-X)
    Obviously for X >= Z, we always win, since Y cannot be less than zero.  If Z-X>N, we always lose, since Y cannot be larger than N.  For anything in between, we have the probability N-1[N-(Z-X)+1].  For example say we were trying to make a DC 15 check with a +5 modifier.  We'd need to roll 10 or better to make it.  That means there's 11 different rolls which would mean success.  Hence the equation.  Now, if a 20 automatically succeeds, as with saves, then the minimum probability of success is 5%.  If 1 always fails, as with saves, then the maximum probability of success is 95%.
    [/SBLOCK]
    Formulas:
    If your modifier is smaller than the DC: Probability of success = N-1[N-(Z-X)+1]
    If your modifier is equal to or greater than the DC: 100% success, unless 1 fails, then 95%.
    If your modifier is N+1 less than the DC: 0% success, unless 20 is auto-success, then 5%.
  • Step 4: Weighted Damage (if applicable): If your character deals damage, then it needs to be weighted by its probability of success to determine just how good it is.  One can say, "assuming all hits" and come up with a huge damage figure, but it means nothing if you have only a 1% chance of landing all hits.  Calculating weighted damage is as simple as multiplying the probability of each attack by the damage that it deals.
  • Probability Grading: If you want to "certify" your character or give it some "grade" of performance, I suggest the following:
    • Passable: 50% probability of success for binary attack forms (like save or die spells).  10% of total monster HP in weighted damage dealt.  With 4 passable party members, 10% weighted damage means that the monster will go down in a little over 2 rounds.  Provided the monster is also passable, this should be alright.  These probabilities are calculated from the AVERAGE values for CR = level +1.
    • Optimal: 75% probability of success for binary attack forms (like save or die spells).  20% of total monster HP in weighted damage dealt.  With 4 passable party members, 20% weighted damage means that the monster will go down in a little over 1 round.  Provided the monster is also optimal, this should be alright.  These probabilities are calculated from the AVERAGE values for CR = level +2 OR the max values of CR = level.
    • Overpowered: 95% probability of success for binary attack forms (like save or die spells).  50% of total monster HP in weighted damage dealt.  With 4 passable party members, 50% weighted damage means that the monster will go down in under 1 round.  Provided the monster is also overpowered this will be the only way to avoid PK.  These probabilities are calculated from the AVERAGE values for CR = level +3 OR the max values of CR = level +1.
    • Twink: 95% probability of success for binary attack forms (like save or die spells).  100% of total monster HP in weighted damage dealt.  With 4 passable party members, 100% weighted damage means that the monster will go down in a little over 1 rounds.  Provided the monster is also ridiculous, you basically need to one-shot it or it will one-shot you.  These probabilities are calculated from the AVERAGE values for CR = level +5 OR the max values of CR = level +2.
  • Step 3 (Defensive Capabilities): This is pretty simple.  Just calculate the offensive capabilities of your monster, like you calculated yours, find his probability of success.  Your probability of successfully defending yourself is 1 - monster's probability of success.  Some considerations:
    • AC versus multiple opponents: The most optimal attack method for a group of monsters fighting a group of characters of higher level is to focus fire on the most damaging or threatening individuals.  Essentially you need to determine if you would be a threatening target to the monsters.  Estimate how many of them you would attract fire from.  If you plan on being the party "tank" assume that all will attack you, as a worst case scenario analysis.  Calculate the probability that each one will hit you (as with the single enemy case) then do the following:
      • The probability that N-N (zero) monsters of N total monsters will hit you is equal to the product of your probability of success against each individual monster.
      • The probability that N - m, where m < N, will hit you is equal to the product your probability of success for the m that miss you, multiplied by the product of your AC's probability of failure for the N- m monsters that hit you.  Your probability of failure = 100% - Probability of Success.
    • Miss Chances: For miss chances you simply multiply the probability of the attacker's success by the percentage and then use this modified offensive success rate to calculate your defense rate (as discussed above).  If multiple miss chances are present take the product of them all and then multiply that by the attacker's success rate.
Interesting Observations on the Data:
This will be an expanding list of observations that I make on the data.  There is no real overarching theme for these observations as of yet.  Just interesting trends.
Hit point Trends:
  • Average HP from CR 0-15: HP scale very linearly in this region at about 12.5 HP per CR.
  • Average HP from CR 15-66: HP scale linearly (I can't tell you the chi-squared of the fit without additional data) at about a rate of 49.6 HP per CR minus 647.
This tells me that damaging characters need to linearly increase their damage output with level and at about level 14 they need to receive a significant damage boost to enter the next linear regime.

Average Saving Throws:
Saving throws increase linearly in the region of 1-30 CR, which is the most interesting area.  Fort saves are generally the highest, with will saves being the lowest.  Interestingly though, ref saves increase at a decreased rate compared to will and fort, such that near 17-23 CR there's a region where Ref saves are lowest on average.  That's good to know for AoE mages.

20th level Average Saves:
Avg saves are: 22.5 fort, 16 ref, 19.12 will.  Lets assume a character with 18 starting int/cha/wis, +5 levels, +5 tome, +6 item and +2 DC from feats casting a 9th level spell with binary save.  Spell DC is 19+12+2 = 33.  Against fort save that gives the monster about a 50% chance of survival, against ref its 20%, and against will its about a 35%.  Assuming the spell slays the monster outright, it's equivalent of dealing 100% avg HP dmg (weighted with the probability of monster's failure).  At 20 CR this is 364 dmg.  Best case is ref, and against a single creature it deals 291.2 weighted damage.  Of course most ref-saves are area effects and thus would apply to more than one creature.  Say it was against 4 lvl 16 creatures (a EL 20 encounter), slaying them outright with one ref-save spell of 9th level would yield 778 weighted damage.
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Re: Optimization by the numbers.
« Reply #21 on: March 01, 2011, 08:41:12 PM »
Er, maybe I missed something. Why did you quote the OP's entirety for no apparent reason?
So you end up stuck in an endless loop, unable to act, forever.

In retrospect, much like Keanu Reeves.

Ithamar

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Re: Optimization by the numbers.
« Reply #22 on: March 01, 2011, 08:48:39 PM »
Because he's afraid Sunic will delete the post probably.
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Anarchy_Kanya

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Re: Optimization by the numbers.
« Reply #23 on: March 01, 2011, 09:04:39 PM »
@ lans
Good thinking. :clap
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[spoiler]
Once my 12 level VoP exalted Monk beat a Horned Devil. ALONE! The only dmg the devil could do to me was from one spell-like that he menaged to shoot out before he droped. :D
wow... you are the epitome of trolling...

Non complete list of people here that are not fuckwits: bearsarebrown, BeholderSlayer, Dictum Mortum, Ithamar, Kajhera, RelentlessImp, SeekingKnight, Shiki, Solo.
You didn't include yourself. Does that mean you're a fuckwit? ???

Hi Welcome
Then I can assume the answer to the question is "Yes".
[/spoiler]
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Ithamar

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Re: Optimization by the numbers.
« Reply #24 on: March 01, 2011, 09:26:25 PM »
Well considering it isn't even Sunic's work product, I don't really see the need personally. *shrug*
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Bauglir

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Re: Optimization by the numbers.
« Reply #25 on: March 01, 2011, 09:46:09 PM »
Seems like a better approach would be to copypaste it into a document on your own computer, or something, rather than provoking him to do so. Bit of a dick move, is what I'm saying. Much as I disagree with Sunic, repeatedly and at length, he doesn't seem to be the "Fuck you, I'm taking my ball with me" type of asshole.
So you end up stuck in an endless loop, unable to act, forever.

In retrospect, much like Keanu Reeves.

Anarchy_Kanya

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Re: Optimization by the numbers.
« Reply #26 on: March 01, 2011, 09:49:35 PM »
But he did it once. He could do it again.
Fly Away With Me!
We Are Angels! Anarchy!
[spoiler]
Once my 12 level VoP exalted Monk beat a Horned Devil. ALONE! The only dmg the devil could do to me was from one spell-like that he menaged to shoot out before he droped. :D
wow... you are the epitome of trolling...

Non complete list of people here that are not fuckwits: bearsarebrown, BeholderSlayer, Dictum Mortum, Ithamar, Kajhera, RelentlessImp, SeekingKnight, Shiki, Solo.
You didn't include yourself. Does that mean you're a fuckwit? ???

Hi Welcome
Then I can assume the answer to the question is "Yes".
[/spoiler]
Making Fighters special - lil project by me.

Bauglir

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Re: Optimization by the numbers.
« Reply #27 on: March 01, 2011, 09:54:23 PM »
Oh, I missed that, then. In that case, consider my judgment of Sunic's character revoked, and my objection revised to "Why remind him?"
So you end up stuck in an endless loop, unable to act, forever.

In retrospect, much like Keanu Reeves.

RelentlessImp

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Re: Optimization by the numbers.
« Reply #28 on: March 01, 2011, 09:56:22 PM »
Oh, I missed that, then. In that case, consider my judgment of Sunic's character revoked, and my objection revised to "Why remind him?"

Don't mind Kanya. She's just been trollbaiting and flaming people from her very first post here without contributing much overall.
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I prepared irresistible phantasmal killer today... as a 4th level spell.
"He hit me with a flaming garbage can...and then the god damn hobo next to it!"

Anarchy_Kanya

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Re: Optimization by the numbers.
« Reply #29 on: March 01, 2011, 10:01:17 PM »
Oh, I missed that, then. In that case, consider my judgment of Sunic's character revoked, and my objection revised to "Why remind him?"

Don't mind Kanya. She's just been trollbaiting and flaming people from her very first post here without contributing much overall.
Yes, that is true and now I'm trying to stop with it. Notice how I'm not flailing at Sunic, or anything like that.
Of course you don't have to change your opinion about me. I realy don't care what HE or YOU think of me.
Fly Away With Me!
We Are Angels! Anarchy!
[spoiler]
Once my 12 level VoP exalted Monk beat a Horned Devil. ALONE! The only dmg the devil could do to me was from one spell-like that he menaged to shoot out before he droped. :D
wow... you are the epitome of trolling...

Non complete list of people here that are not fuckwits: bearsarebrown, BeholderSlayer, Dictum Mortum, Ithamar, Kajhera, RelentlessImp, SeekingKnight, Shiki, Solo.
You didn't include yourself. Does that mean you're a fuckwit? ???

Hi Welcome
Then I can assume the answer to the question is "Yes".
[/spoiler]
Making Fighters special - lil project by me.

BeholderSlayer

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Re: Optimization by the numbers.
« Reply #30 on: March 01, 2011, 10:07:09 PM »
« Last Edit: March 01, 2011, 10:10:58 PM by BeholderSlayer »
Hi Welcome
[spoiler]
Allow me to welcome you both with my literal words and with an active display of how much you fit in by being tone deaf, dumb, and uncritical of your babbling myself.[/spoiler]

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I prepared irresistible phantasmal killer today... as a 4th level spell.
"He hit me with a flaming garbage can...and then the god damn hobo next to it!"

Bauglir

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Re: Optimization by the numbers.
« Reply #32 on: March 01, 2011, 10:21:56 PM »
Oh, I missed that, then. In that case, consider my judgment of Sunic's character revoked, and my objection revised to "Why remind him?"

Don't mind Kanya. She's just been trollbaiting and flaming people from her very first post here without contributing much overall.

From what's left of that thread, it looks like Sunic ragequitted too early. There were occasional bursts of people being dicks for no good reason, but he started them as often as not, and there seemed to be an actual discussion going on with, y'know arguments. That said, there appear to be deletions that I'm missing, so I'm not really getting a complete picture; I'll give Sunic the benefit of the doubt, here, and assume somebody started dragging their issues from another thread into that one. I wouldn't put it past some of the posters here, and while Sunic isn't one of the ones immune to that particular form of jackassery, I doubt he'd start something like this and then delete his own material. And he's not the sort to blatantly lie (I saw no warning, so I assume it was deleted as part of a larger post that needed to go), either, even if I think he's mistaken in a lot of mechanics areas.

Point being, sounds like people were dragging baggage into a thread where it didn't belong and was genuinely useful (from what I recall of the OP when it was there), because they have a grudge against Sunic. I mean, guys, it doesn't matter how right you are about something. Don't be a giant annoying douchebag about it. I mean, that's 90% of the reason I end up arguing with Sunic in the first place, don't make me take his side on that same fucking issue. And now they're dragging that baggage into this one? Real classy. Seriously, stop whining. It's gone. If you're afraid something's going to be deleted, make backups, but don't be an ass about it. You're not helping your chances of keeping it here, dumbass.

EDIT: Also, dammit guys. Now we're going to have to worry about jewel thieves. SEE WHAT YOU'VE DONE.
So you end up stuck in an endless loop, unable to act, forever.

In retrospect, much like Keanu Reeves.

Lycanthromancer

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Re: Optimization by the numbers.
« Reply #33 on: March 01, 2011, 10:28:17 PM »
EDIT: Also, dammit guys. Now we're going to have to worry about jewel thieves. SEE WHAT YOU'VE DONE.
Avoid veterinarians.
[spoiler]Masculine men like masculine things. Masculine men are masculine. Therefore, liking masculine men is masculine.

I dare anyone to find a hole in that logic.
______________________________________
[/spoiler]I'm a writer. These are my stories. Some are even SFW! (Warning: Mostly Gay.)
My awesome poster collection. (Warning, some are NSFW.)
Agita's awesome poster collection.
[spoiler]
+1 Lycanthromancer
Which book is Lycanthromancer in?
Lyca ... is in the book. Yes he is.
 :D
shit.. concerning psionics optimization, lycan IS the book
[/spoiler]

RelentlessImp

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Re: Optimization by the numbers.
« Reply #34 on: March 01, 2011, 10:28:38 PM »
Oh, I missed that, then. In that case, consider my judgment of Sunic's character revoked, and my objection revised to "Why remind him?"

Don't mind Kanya. She's just been trollbaiting and flaming people from her very first post here without contributing much overall.

From what's left of that thread, it looks like Sunic ragequitted too early. There were occasional bursts of people being dicks for no good reason, but he started them as often as not, and there seemed to be an actual discussion going on with, y'know arguments. That said, there appear to be deletions that I'm missing, so I'm not really getting a complete picture; I'll give Sunic the benefit of the doubt, here, and assume somebody started dragging their issues from another thread into that one. I wouldn't put it past some of the posters here, and while Sunic isn't one of the ones immune to that particular form of jackassery, I doubt he'd start something like this and then delete his own material. And he's not the sort to blatantly lie (I saw no warning, so I assume it was deleted as part of a larger post that needed to go), either, even if I think he's mistaken in a lot of mechanics areas.

Point being, sounds like people were dragging baggage into a thread where it didn't belong and was genuinely useful (from what I recall of the OP when it was there), because they have a grudge against Sunic. I mean, guys, it doesn't matter how right you are about something. Don't be a giant annoying douchebag about it. I mean, that's 90% of the reason I end up arguing with Sunic in the first place, don't make me take his side on that same fucking issue. And now they're dragging that baggage into this one? Real classy. Seriously, stop whining. It's gone. If you're afraid something's going to be deleted, make backups, but don't be an ass about it. You're not helping your chances of keeping it here, dumbass.
rs
EDIT: Also, dammit guys. Now we're going to have to worry about jewel thieves. SEE WHAT YOU'VE DONE.

There was a thread deleted by Sunic because people were doing exactly that. And now people are basically bitching about it, despite them being the reason the thread had to go (dragging in all the bitching and flailfests and whining from other threads into it). So now they're taking cheap shots at someone who can't defend themselves due to a temporary suspension.

So basically, :fu to pretty much everyone doing that, it's in shitty taste and bad form.

In other news, DeBeers Corporation has hired assassins to take out competitors selling diamonds cheaper than them. Discuss.
Solo on Essentials
I prepared irresistible phantasmal killer today... as a 4th level spell.
"He hit me with a flaming garbage can...and then the god damn hobo next to it!"

Lycanthromancer

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Re: Optimization by the numbers.
« Reply #35 on: March 01, 2011, 10:37:10 PM »
There was a thread deleted by Sunic because people were doing exactly that. And now people are basically bitching about it, despite them being the reason the thread had to go (dragging in all the bitching and flailfests and whining from other threads into it). So now they're taking cheap shots at someone who can't defend themselves due to a temporary suspension.

So basically, :fu to pretty much everyone doing that, it's in shitty taste and bad form.

In other news, DeBeers Corporation has hired assassins to take out competitors selling diamonds cheaper than them. Discuss.
Screaming 'HI WELCOME' repeatedly and calling people fuckwits is hardly defending yourself.
[spoiler]Masculine men like masculine things. Masculine men are masculine. Therefore, liking masculine men is masculine.

I dare anyone to find a hole in that logic.
______________________________________
[/spoiler]I'm a writer. These are my stories. Some are even SFW! (Warning: Mostly Gay.)
My awesome poster collection. (Warning, some are NSFW.)
Agita's awesome poster collection.
[spoiler]
+1 Lycanthromancer
Which book is Lycanthromancer in?
Lyca ... is in the book. Yes he is.
 :D
shit.. concerning psionics optimization, lycan IS the book
[/spoiler]

RelentlessImp

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Re: Optimization by the numbers.
« Reply #36 on: March 01, 2011, 10:39:51 PM »
There was a thread deleted by Sunic because people were doing exactly that. And now people are basically bitching about it, despite them being the reason the thread had to go (dragging in all the bitching and flailfests and whining from other threads into it). So now they're taking cheap shots at someone who can't defend themselves due to a temporary suspension.

So basically, :fu to pretty much everyone doing that, it's in shitty taste and bad form.

In other news, DeBeers Corporation has hired assassins to take out competitors selling diamonds cheaper than them. Discuss.
Screaming 'HI WELCOME' repeatedly and calling people fuckwits is hardly defending yourself.

I didn't say it was a valid defense. I just said it was a defense. It's more of one than not getting to respond at all to slanderous claims and bullshit posts.
Solo on Essentials
I prepared irresistible phantasmal killer today... as a 4th level spell.
"He hit me with a flaming garbage can...and then the god damn hobo next to it!"

Anarchy_Kanya

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Re: Optimization by the numbers.
« Reply #37 on: March 01, 2011, 10:44:48 PM »
Oh, I missed that, then. In that case, consider my judgment of Sunic's character revoked, and my objection revised to "Why remind him?"

Don't mind Kanya. She's just been trollbaiting and flaming people from her very first post here without contributing much overall.

From what's left of that thread, it looks like Sunic ragequitted too early. There were occasional bursts of people being dicks for no good reason, but he started them as often as not, and there seemed to be an actual discussion going on with, y'know arguments. That said, there appear to be deletions that I'm missing, so I'm not really getting a complete picture; I'll give Sunic the benefit of the doubt, here, and assume somebody started dragging their issues from another thread into that one. I wouldn't put it past some of the posters here, and while Sunic isn't one of the ones immune to that particular form of jackassery, I doubt he'd start something like this and then delete his own material. And he's not the sort to blatantly lie (I saw no warning, so I assume it was deleted as part of a larger post that needed to go), either, even if I think he's mistaken in a lot of mechanics areas.

Point being, sounds like people were dragging baggage into a thread where it didn't belong and was genuinely useful (from what I recall of the OP when it was there), because they have a grudge against Sunic. I mean, guys, it doesn't matter how right you are about something. Don't be a giant annoying douchebag about it. I mean, that's 90% of the reason I end up arguing with Sunic in the first place, don't make me take his side on that same fucking issue. And now they're dragging that baggage into this one? Real classy. Seriously, stop whining. It's gone. If you're afraid something's going to be deleted, make backups, but don't be an ass about it. You're not helping your chances of keeping it here, dumbass.
rs
EDIT: Also, dammit guys. Now we're going to have to worry about jewel thieves. SEE WHAT YOU'VE DONE.

There was a thread deleted by Sunic because people were doing exactly that. And now people are basically bitching about it, despite them being the reason the thread had to go (dragging in all the bitching and flailfests and whining from other threads into it). So now they're taking cheap shots at someone who can't defend themselves due to a temporary suspension.

So basically, :fu to pretty much everyone doing that, it's in shitty taste and bad form.

In other news, DeBeers Corporation has hired assassins to take out competitors selling diamonds cheaper than them. Discuss.
I didn't start it.
And BTW: Sunic got suspended? Like in "temporary banned"? O_o Wierd. Does the moderation actualy do their job in here?
Fly Away With Me!
We Are Angels! Anarchy!
[spoiler]
Once my 12 level VoP exalted Monk beat a Horned Devil. ALONE! The only dmg the devil could do to me was from one spell-like that he menaged to shoot out before he droped. :D
wow... you are the epitome of trolling...

Non complete list of people here that are not fuckwits: bearsarebrown, BeholderSlayer, Dictum Mortum, Ithamar, Kajhera, RelentlessImp, SeekingKnight, Shiki, Solo.
You didn't include yourself. Does that mean you're a fuckwit? ???

Hi Welcome
Then I can assume the answer to the question is "Yes".
[/spoiler]
Making Fighters special - lil project by me.

Bozwevial

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Re: Optimization by the numbers.
« Reply #38 on: March 01, 2011, 10:48:31 PM »
And BTW: Sunic got suspended? Like in "temporary banned"? O_o Wierd. Does the moderation actualy do their job in here?
The mods tend to take an active role only when things go beyond a certain point. They're pretty prompt about it, so yes, they do monitor the boards.

Bauglir

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Re: Optimization by the numbers.
« Reply #39 on: March 01, 2011, 10:49:32 PM »
There was a thread deleted by Sunic because people were doing exactly that. And now people are basically bitching about it, despite them being the reason the thread had to go (dragging in all the bitching and flailfests and whining from other threads into it). So now they're taking cheap shots at someone who can't defend themselves due to a temporary suspension.

So basically, :fu to pretty much everyone doing that, it's in shitty taste and bad form.

In other news, DeBeers Corporation has hired assassins to take out competitors selling diamonds cheaper than them. Discuss.
Screaming 'HI WELCOME' repeatedly and calling people fuckwits is hardly defending yourself.

This is true, but apparently he's been suspended and thus literally cannot defend himself? If I understand correctly.

Besides, the point is that you should respond to him when he does that, and only when he does that. He hasn't in this thread, so don't bring it up. Don't drag baggage from other threads in. If you have a problem with him that you need to address, and it doesn't really fit in another thread, START ONE. It's not that hard.

Sure, maybe he's too much of an ass to back down, but that doesn't make you any less of an ass if you don't either. If you think he's a dick, then don't sink to his level. Don't destroy threads because you're too prideful to move your discussion somewhere appropriate.

I can't repeat this enough (and yes, I did steal the line from somewhere else, doesn't make it any less applicable); it doesn't matter how right you are. If you're a giant annoying douchebag about it, you're still a giant annoying douchebag. Shut up and quit making your side look bad.
So you end up stuck in an endless loop, unable to act, forever.

In retrospect, much like Keanu Reeves.