To compare these two techniques you can calculate damage per round:
dpr = % chance to hit x average damage
Let's assume you're an OD&D 4th level fighter needing a 10 to hit leather (AC 7). That's 55% chance. Now if you roll two dice and take the best, that's the same as saying 'What is my chance of hitting at least once on two dice?'.
This is easier to calculate if you reverse the question: 'What is the chance that I not hit with two dice?'. This is %miss x %miss = 45% x 45% = 20.25%
So your chance to hit is 1 - 0.2025 = 0.7975
And average damage on a d6 is 3.5 so:
DPR for roll atk twice = 0.7975 x 3.5 = 2.79125 dmg per round
Not sure I'm following all this, but average damage per round is increasing from 1.9 (.55 x 3.5) to 2.79 (.7975 x 3.5)? Isn't that the same as giving the fighter almost a +5 to hit? That seems like a large to-hit bonus for OD&D.
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Re: Fuzzy on the statistics... « Reply #16 on Feb 15, 2012, 11:11pm »
It might not be obvious to folks from the above that X2 is the higher of the two dice rolls.
Yes, thanks for pointing this out. I didn't explain the notation in my original post. So here's more detail about the notation used in order statistics:
X1,...,Xn are the unordered/unsorted random variables (also called variates) x1,...,xn are the unordered/unsorted observations. Notice upper-case for variates. Lower-case letters for observations. X(1) <= ... <= X(n) are the ordered/sorted variates, called the "order statistics." Notice the parentheses around the subscripts. x(1) <= ... <= x(n) are the ordered observations. Again, observations are lower-case. Random variables (or variates) are upper case. Parentheses for order.
Some authors prefer this notation for the order statistics: X1:n <= ... <= Xn:n, called "extensive form." No parentheses in the subscripts, but the sample size "n" is shown explicitly.
As harbinger said, this stuff is trickier. Hope this makes it a little clearer.
Not sure I'm following all this, but average damage per round is increasing from 1.9 (.55 x 3.5) to 2.79 (.7975 x 3.5)? Isn't that the same as giving the fighter almost a +5 to hit? That seems like a large to-hit bonus for OD&D.
Good point. This is why AD&D two-weapon fighting is given a -2 on the main weapon and -4 on the off-hand weapon (I think those were the numbers). I've read somewhere that evens out the bonus. Let's see with a d8 sword and a d6 short sword.
.55 x 4.5 = 2.475
.45 x 4.5 + .35 x 3.5 = 3.25
Nope, it's still a +3 ~ +4
-6 and -4 works:
.35 x 4.5 + .25 x 3.5 = 2.45
Of course, it's still a curve (not linear), so this only matches at this particular character level.
Of course, it's still a curve (not linear), so this only matches at this particular character level.
Mmhmm! I managed to create the same graph you made earlier with the basic single weapon DPR in as a base-line. Though it occurs to me that I have no idea how to upload it as an attachment, maybe I should just upload it to a separate imagehost.
I think I am actually a big fan of how this is falling out so that there are strong situational drives to certain fighting styles. I think that once it is more fleshed out it will only get better. Remember that while using that second hand for either a second weapon or a double grip on a single weapon will always net a basic DPR advantage it can hinder your character in other ways.
Of course I tend to lean heavily towards tracking things like weight and logistics, so it becomes a pertinent questions in a number of situations whether you have a free hand for balance as you fight on a cliff face, will you have to drop your weapon to save yourself when a trap goes off, do you really want the extra encumbrance of lugging around another sword, etc. E.g. in my last session my party ran into a classic trapdoor that threatened to dump them two levels down into a lake housing a small shrine with a water elemental. Those with free hands got a chance to grab onto something and save themselves, those with full hands had to choose between taking the plunge and dropping whatever they were holding into the water below, and once they hit the water we began looking at just how much weight was dragging them to a watery grave and so wound up with characters desperately dropping their weapons and packs to try to swim.
And this only becomes more interesting if you introduce weapons with damages beyond 1d6. I might try to run those numbers.
Re: Fuzzy on the statistics... « Reply #19 on Feb 20, 2012, 10:20am »
Okay, so I have been running more numbers and I stumbled across something really interesting!
As we all know the average result of rolling 1d6 is 3.5. Likewise the average result on 1d4 is 2.5 and on 1d8 it is 4.5. So here's the thing: the average when rolling 2d6 and keeping the highest is 4.47 or ~4.5. And the average result when rolling 2d6 and keeping the lowest is 2.52. So basically, averaged over time, you get almost identical results from forcing a player to roll 1d6 twice and take the max or min as you would from scaling the die type. Which I think is cool.
Okay, so I have been running more numbers and I stumbled across something really interesting!
As we all know the average result of rolling 1d6 is 3.5. Likewise the average result on 1d4 is 2.5 and on 1d8 it is 4.5. So here's the thing: the average when rolling 2d6 and keeping the highest is 4.47 or ~4.5. And the average result when rolling 2d6 and keeping the lowest is 2.52. So basically, averaged over time, you get almost identical results from forcing a player to roll 1d6 twice and take the max or min as you would from scaling the die type. Which I think is cool.
Welcome to the world of neat'o mechanics!
I've always liked using: - Magic User - take min of 2d6 - Cleric - d6 - Fighter - take max of 2d6
Or you could say 'light', 'medium', 'martial' weapon.
So basically, averaged over time, you get almost identical results from forcing a player to roll 1d6 twice and take the max or min as you would from scaling the die type.
In the long run, for large samples, the differences between the sample means of d8 and max{d6,d6} may be negligible. However, in the short run, for small samples, the differences between these two distributions could be critical.
Let's say I get one last shot at a foe before I'm paralyzed. I make the hit. Now I'm about to roll for damage.
Case (a) I need to roll 7 or more, or I'm doomed for sure. My chances are 0.25 on d8, but zero on max{d6,d6}. Case (b) I need 5 or more. The probability of rolling 5 or more on d8 is 0.5. The probability of rolling 5 or more on max{d6,d6} is 0.555 (5 repeats forever), giving me a slight edge (0.0555...) over d8. Case (c) The only thing I can't roll is a "1". The probability of rolling a "1" is 1/8 on a d8 but only 1/36 on a max{d6,d6}. So the chance of rolling "1" on a d8 is 4.5 times higher on a d8 than on max{d6,d6}.
The word "average" is somewhat ambiguous. While it commonly refers to "mean", it could mean...
mean: expected value E(X), i.e., sum(x f(x) for each outcome x) median: middle value m; F(m) >= 1/2 and (1 - F(m) + f(m)) >= 1/2 mode: most frequent value; x for which f(x) attains its maximum (where f is the pmf and F is the cdf).
With this in mind, the "averages" of these two distributions -- d8 and max{d6,d6} -- are somewhat different:
mean of d8 = 4.5 ≈ mean of max{d6,d6} = 4.47222 (2 repeats forever) median d8 = 4.5, but median max{d6,d6} = 5 d8 is modeless, but mode of max{d6,d6} is 6
Furthermore, the spreads of these two distributions are quite different:
This means a random sample drawn from max{d6,d6} will be clustered together more closely around the mean, than will a random sample drawn from d8.
Here's a summary of the properties of all the distributions under consideration in this thread:
d4 mean = E(d4) = 5/2 = 2.5 median = 2.5 mode = undefined variance = var(d4) = 5/4 = 1.25 stddev = sqrt(5)/2 ≈ 1.11803 pmf = P(d4=x) = 1/4 for x in {1,2,3,4}; 0 otherwise cdf = P(d4<=x) = floor(x)/4 for x in [0,4] N.B. This is a non-decreasing, right-continuous step function
min{d6,d6} mean = 91/36 = 2.52778 median = 2 mode = 1 var = 2555/1296 = 1.9714506172839506172839 (506172839 repeating) stddev = sqrt(2555)/36 ≈ 1.40408355067779191 pmf = P(min{d6,d6}=x) = (13-2x)/36 for x in {1,2,3,4,5,6}; 0 otherwise cdf = P(min{d6,d6}<=x) = (12x - x^2)/36 after step(6) where "after" means function composition and "step(6)" is a step function, defined below.
d6 mean = 7/2 = 3.5 median = 3.5 mode = undefined var = 35/12 = 2.91666 (6 repeats forever) stddev ≈ 1.707825 pmf = P(d6=x) = 1/6 for x in {1,2,3,4,5,6}; 0 otherwise cdf = P(d6<=x) = floor(x)/6 for x in [0,6]
max{d6,d6} mean = 161/36 = 4.472222 (2 repeats forever) median = 5 mode = 6 var = 2555/1296 = 1.9714506172839506172839 (506172839 repeating) stddev = sqrt(2555)/36 ≈ 1.40408355067779191 pmf = P(min{d6,d6}=x) = (2x-1)/36 for x in {1,2,3,4,5,6}; 0 otherwise cdf = P(min{d6,d6}<=x) = (x^2)/36 after step(6)
d8 mean = 9/2 = 4.5 median = 4.5 mode = undefined var = 21/4 = 5.25 stddev = sqrt(21)/2 ≈ 2.29129 pmf = P(d8=x) = 1/8 for x in {1,2,3,4,5,6,7,8}; 0 otherwise cdf = P(d8<=x) = floor(x)/8 for x in [0,8]
Where...
step(a) : Reals -> {0,1,2,...,a} is a step function, defined as step(a) = 0 χ(-∞,1) + sum(b χ[b,b+1) for b in {1,...,a-1}) + a χ[a,∞)
χA : Reals -> {0,1} is the indicator function of the set A, defined as χA(x) = 1 if x in A; 0 otherwise
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I didn't explain the notation in my original post. So here's more detail about the notation used in order statistics:
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