Seeking Math Wizard for Weapon Mastery

I've been trying to work out the math for weapon mastery, and it's beyond my mathematical abilities. I can do it for 2 d m drop the lowest, but not above that for when you have Deadly Strike.

Can anyone explain how to create a sum E for m d n dice drop the lowest?

(for 1d12, weapon mastery increases your average damage from 6.5 to 8.49.)

I don't think you really want to do that, but ...

What changes for drop the lowest is that each die can now take on the value of 0 as a result (ie: be the dropped die). Further, the probability of a die being zero is both a function of the initial roll result and the result of every other die. So now two things have changed with respect to a normal die averaging:

1) all results for a single die are no longer equal, and much worse, 

2) the result of a single die depends on the results of every other die (you must calculate the probability that each die result is the lowest die result of all of the dice in the throw).

Someone smarter may come up with something better, but I don't know a better way than explicitly calculating the whole range of possible outcomes and directly calculating the probabilities from that. The range of possible outcomes is (die size)^(number of die), For weapon master and 5d12 Deadly strike, that is 12^6 or
2985984 possible states.

These are just monte carlo results. I don't remember is it is 1 million or 10 million throws each, but it should be pretty close the true averages. You'll note that the damage increase scales as the square root of the increase in the number of dice, and just a hair better than linearly with the die size. The relative advantage of martial expertise falls off significantly with more dice. It actaully retains its value better n X 3 classes vs the n X 5 classes.





















































































































































































































Die SizenDicew/o Expertisew/ExpertiseIncrease% Increase
412.503.120.6224.91%
425.005.940.9418.77%
437.508.611.1214.91%
4410.0011.231.2312.30%
4512.5013.811.3110.45%
613.504.470.9727.76%
627.008.451.4620.84%
6310.5012.251.7416.60%
6414.0015.931.9313.79%
6517.4919.552.0611.78%
814.505.811.3129.21%
829.0010.971.9721.87%
8313.4915.852.3617.50%
8418.0020.612.6114.52%
8522.4925.292.8012.43%
1015.507.151.6529.95%
10211.0113.482.4722.47%
10316.5019.462.9717.97%
10422.0025.293.2914.96%
10527.5031.023.5212.80%
1216.518.491.9830.48%
12213.0015.972.9722.89%
12319.5023.073.5718.33%
12425.9929.963.9715.27%
12532.5036.754.2413.06%






www.anydice.com

output [highest 5 of 6d6] 

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

Looks like you could just do a flat bonus of 1/4 the Weapon die rounded down if you didn't want to do all the extra die rolling every turn.

d4 and d6: +1 damage 

d8 and d10: +2 damage

d12: +3 damage

It would be a larger add at the low levels and less at the higher but it would be close.         
Thanks guys. I just found anydice.com and came here to post my success in finding an answer...but you got there first.

I still want to know how to actually calculate the probability of any given result for any given n d m, just for my own personal betterment, but the answer will suffice for now.
With respect to yakuba, I don't understand his idea that you can treat a roll as zero. Here's how I understand it.

You're going to roll N dice, each D-sided. The average total is N*(D+1)/2. You'll drop the lowest, so you need the average roll for the lowest die. Call the lowest die's value d_min.

What's the probability that d_min is greater than or equal to 1? Obviously one, since it can't be less than one. Call that P_1+.
So P_1+ = 1.

What's the probability that d_min is greater than or equal to 2? That happens if none of your N dice roll a 1. For each die, there are D-1 ways to not roll a 1, so each die has a probability of (D-1)/D to roll above a 1. The N rolls are independent, so the probability that d_min >= 2 is
P_2+ = [(D-1)/D]^N

What's the prob that d_min >= 3? Now there are D-2 ways to do that for each die, so the prob per die is (D-2)/D, and the prob for all N dice is
P_3+ = [(D-2)/D]^N

You can see the pattern. The probability for d_min >= d is
P_d+ = [(D-d+1)/D]^N
That works even for d_min = D, since the only way that happens is if all N dice roll their max, which has a probability of 1/D^N.

So now what's the probability that d_min is equal to 1? That the same as the prob that it is greater than or equal to 1, minus the prob that it is greater than or equal to 2. So you can say
P_1 = (P_1+) - (P_2+)
and in general,
P_d = (P_d+) - (P_(d+1)+)
That still works for d=D since P_(D+1)+ = 0.

Now by definition, the average value for d_min will be the sum of d*P_d over all possible d's:
(avg d_min) = 1*P_1 + 2*P_2 + 3*P_3 + ... + D*P_D
  = 1*(P_1+ - P_2+) + 2*(P_2+-P_3+) + ... + D*(P_D+)
If you rearrange the terms, this is
= (P_1+)*1 + (P_2+)*(2-1) + (P_3+)*(3-2) + ... + (P_D+)*[D-(D-1)]
  = (P_1+) + (P_2+) + ... + (P_D+)
And you can write that simply as
(avg d_min) = sum_d (d/D)^N

You can't simplify that any further, but it gives you a formula. So for instance if D = 6 and N = 2, you'd have
(avg d_min) = (1/6)^2 + (2/6)^2 + (3/6)^2 + (4/6)^2 + (5/6)^2 + (6/6)^2
= 2.53

So if you roll 2d6 and drop the lowest, your average result will be 7-2.53 = 4.47, which agrees with the Monte Carlo yakuba did.


Well, you asked!

With respect to yakuba, I don't understand his idea that you can treat a roll as zero. Here's how I understand it.




I like that! Thanks, I did not see that way of thinking of the problem. I tend to lean towards the brute force methodologies.

As for what I meant, consider the simplist case of 2d2
The actual rolls are: (1,1),(1,2),(2,1),(2,2). But the actual results are simply (1),(2),(2),(2) which I was thinking of as
(1,0),(0,2),2,0),(2,0). It's a far less elegant way of looking at the problem, I wouldn't worry about it. 

I like that! Thanks, I did not see that way of thinking of the problem. I tend to lean towards the brute force methodologies.


Cool. I guess one thing I might add, for brute-forcing a problem like this it's probably easier to just enumerate all the possibilities rather than doing a real Monte Carlo. Even rolling six 12-sided dice, there are only 12^6 ~ 3 million different rolls. So if you're doing a Monte Carlo with around a million trials, you could just as easily step through every possible result and get an exact average. And of course, with smaller numbers and dice, it will go much faster. Rolling 2d4 only requires 16 steps.

Cool. I guess one thing I might add, for brute-forcing a problem like this it's probably easier to just enumerate all the possibilities rather than doing a real Monte Carlo. Even rolling six 12-sided dice, there are only 12^6 ~ 3 million different rolls. So if you're doing a Monte Carlo with around a million trials, you could just as easily step through every possible result and get an exact average. And of course, with smaller numbers and dice, it will go much faster. Rolling 2d4 only requires 16 steps.



The number of actual cases is orders of magntiude smaller, but the actual run time is only seconds even for 10 million tries for every condition, and it is much easier and faster to write a script that rolls x and takes x-1.

I'm lazy.


I like that! Thanks, I did not see that way of thinking of the problem. I tend to lean towards the brute force methodologies.


Cool. I guess one thing I might add, for brute-forcing a problem like this it's probably easier to just enumerate all the possibilities rather than doing a real Monte Carlo. Even rolling six 12-sided dice, there are only 12^6 ~ 3 million different rolls. So if you're doing a Monte Carlo with around a million trials, you could just as easily step through every possible result and get an exact average. And of course, with smaller numbers and dice, it will go much faster. Rolling 2d4 only requires 16 steps.

That's how anydice works.

It does every itteration of 2d4.

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.


You're going to roll N dice, each D-sided. 

(avg d_min) = sum_d (d/D)^N

You can't simplify that any further, but it gives you a formula. So for instance if D = 6 and N = 2, you'd have
(avg d_min) = (1/6)^2 + (2/6)^2 + (3/6)^2 + (4/6)^2 + (5/6)^2 + (6/6)^2
= 2.53

So if you roll 2d6 and drop the lowest, your average result will be 7-2.53 = 4.47, which agrees with the Monte Carlo yakuba did.


Well, you asked!


Jaelis, do you know how to calculate a given value? For example, what is the chance of rolling 30 on 6d12 drop the lowest?

I feel like there must be a more elegant way to do this with Binomial Distributions. I've gotten close, but I can't seem to take into account some odd rolls where all the dice have the same value (for example, 2, 2, 2, drop the lowest, this outcome isn't showing up on my binomial distributions).

Your method is a pretty clever way of calculating the average. I should really just take some more college math courses
Your method is a pretty clever way of calculating the average. I should really just take some more college math courses

Why do that when you could just add jaelis to your friends list? ;)

(The secrets to my success revealed!)

Danny

Thanks again Jaelis! It's brilliant!

You're going to roll N dice, each D-sided. Jaelis, do you know how to calculate a given value? For example, what is the chance of rolling 30 on 6d12 drop the lowest?

I feel like there must be a more elegant way to do this with Binomial Distributions. I've gotten close, but I can't seem to take into account some odd rolls where all the dice have the same value (for example, 2, 2, 2, drop the lowest, this outcome isn't showing up on my binomial distributions).

Your method is a pretty clever way of calculating the average. I should really just take some more college math courses


I don't know an elegant way, but this works. You can use the probability generating function to calculate the chance of getting a particular result from a given dice pool. This guy shows you how to do it:
stats.stackexchange.com/questions/3614/h...
and if you want more details, this (and other refs) has the proofs:
www.am.qub.ac.uk/users/g.gribakin/sor/Ch... (eqn 3.17)

So that method can be used, for instance, to find the prob of getting a 36 when rolling 6d12. To handle dropping the lowest, you can use my previous expressions for the probabilty of getting a particular value as the lowest. So the prob to get 30 on 6d12 drop lowest would be
Prob(31)*Prob(1 is lowest) + Prob(32)*Prob(2 is lowest) + etc

But unless you can find a way to simplify that result, it would probably be easier here to just brute force it and add up all the possible results.
I love your right hand rule profile picture by the way (friend of mine pointed it out to me).
I love your right hand rule profile picture by the way (friend of mine pointed it out to me).


Thanks! I teach physics, so it's appropriate.