[Blue Rose]

I would assume that's on a Troll? 

I've finagled it up to 9P[8] -4 AP with four Striking Calluses, Level 2 Bone Density Augmentation (#@$%ing HOBO CHARGEN!) and a weak-ass Qi Focus.

I guess the Qi Focus doesn't affect damage, but it does offset somewhat losing the Weapon focus.

Cyclops is a little better, but yeah.

Troll or better, with striking calluses on both hands and both feet, exceptional strength, genetically optimized strength, strength augmented to max by whatever means, critical strike: unarmed, and maxed bone density (preferably alpha grade) via restricted gear.

Comes out to...
Strength 16.
+2 from striking calluses.
+3 from bone density.
+1 from critical strike.
=21P.

From there, you can add more from martial arts.  Ji Dhao, most notably, is an easy +1.

Options to go up from there?  Cyclops gives you one more point of strength.  HMHVV gives you more strength.  SURGE can give you more strength and can give you more limbs to put striking calluses on.  Elemental Aura does terrible, nasty things to your unarmed strikes.  Or, you could shapechange into an elephant and wreck all of the things.

Okay, since each die only has about a 1/3 chance of being a hit? I guess that does make sense.

In that case, Penetrating Strikes has diminishing returns. Rank 1 gives me a 33.3% chance of nullifying a hit on their damage resist roll.

But R2 is only +22.25%, R3 +14.81%, and 9.88% for R4.

At that rate, I might drop it from R4 to R3, dumping that point back into Combat Sense R3...also diminishing returns, but balanced out vs my current PS 4/CS 2.

-Jn-

No, there really aren't any diminishing returns, unless you're facing incredibly lightly armored enemies.  If you take four dice away from your opponent's soak pool, each individual die had a 1/3 chance of coming up a success.  Or 5/9 if they edge for rerolls.  This is independent of how many dice you take away.

-4 AP is slightly better than +1 1/3 DV.

[JoeNapalm]

Okay, since each die only has about a 1/3 chance of being a hit? I guess that does make sense.

In that case, Penetrating Strikes has diminishing returns. Rank 1 gives me a 33.3% chance of nullifying a hit on their damage resist roll.

But R2 is only +22.25%, R3 +14.81%, and 9.88% for R4.

At that rate, I might drop it from R4 to R3, dumping that point back into Combat Sense R3...also diminishing returns, but balanced out vs my current PS 4/CS 2.

-Jn-

No, there really aren't any diminishing returns, unless you're facing incredibly lightly armored enemies.  If you take four dice away from your opponent's soak pool, each individual die had a 1/3 chance of coming up a success.  Or 5/9 if they edge for rerolls.  This is independent of how many dice you take away.

-4 AP is slightly better than +1 1/3 DV.

How is that not diminishing returns?

The odds of an individual die coming up with a hit is 1-in-3, but you're talking about the odds of each additional die in the cumulative distribution.

Odds of successfully nullifying additional cumulative hits based on Rank:

1 - 33.33%
2 - 22.25%
3 - 14.81%
4 - 9.98%

Each additional Rank costs the same as Rank 1, but drops off in effectiveness.

The more Magic you pump into it, the less it buys you.

Diminishing returns.


-Jn-

[Blue Rose]

How is that not diminishing returns?

The odds of an individual die coming up with a hit is 1-in-3, but you're talking about the odds of each additional die in the cumulative distribution.

Odds of successfully nullifying additional cumulative hits based on Rank:

1 - 33.33%
2 - 22.25%
3 - 14.81%
4 - 9.98%

Each additional Rank costs the same as Rank 1, but drops off in effectiveness.

The more Magic you pump into it, the less it buys you.

Diminishing returns.


-Jn-

I don't know how you're coming up with this math, but it looks like you're reading dice from right to left until you find a single success and then stopping, which is not how AP works.  Your soak pool always applies in its entirety as a penalty to your target's soak pool, so long as your AP doesn't exceed their armor.

Let's say you have no AP.  You are punching Officer Bob, who has Body 3 and Armor 12, for a soak pool of 15.

Each individual die has a 1/3 chance of coming up a success.  Therefore, with fifteen dice, you see an average of 5 successes to soak.

Now, you take rank 1 penetrating strike.  You have AP -1.  You penetrate one point of armor, reducing Officer Bob's soak to 14 dice.

Still, each individual die has a 1/3 chance of success, so with 14 dice, you see an average of 4 2/3 successes to soak, down from 5.  That first rank is worth a third of a point of damage.

Now, you take rank 2 penetrating strike.  You have AP -2.  You penetrate 2 points of armor, reducing Officer Bob's soak to 13 dice.

1/3 chance of success per die, 13 die, average of 4 1/3 successes, down from 4 2/3, so that second rank is worth a third of a point of damage.

Third rank.  AP -3.  Soak pool is 12.

Average soak result of 4, down from 4 1/3.  That third rank is worth a third of a point of damage.

Fourth rank.  AP -4.  Soak is 11.

Average soak is 3 2/3, down from 4.  The fourth rank is also worth a third of a point of damage.  Every single rank is worth a third of a point of damage.  There are no diminishing returns.

If your AP is -4, that means your target is rolling four fewer dice to soak, which means they have four fewer chances to get successes.  They could have come up all successes.  They could have come up all 1's.  But now, they can't, because they aren't being rolled at all.

[Adamo1618]

Every die roll is independent; meaning the outcome of one is unaffected by other dice. Rolling a second die after rolling the first one is not going to affect the outcome of the first. And not vice versa. Hence, every die will have a 1/3 chance to roll a hit. Regardless of the amount of dice.

[JoeNapalm]

Every die roll is independent; meaning the outcome of one is unaffected by other dice. Rolling a second die after rolling the first one is not going to affect the outcome of the first. And not vice versa. Hence, every die will have a 1/3 chance to roll a hit. Regardless of the amount of dice.

Absolutely true...

...if you only care about the outcome of one die.

But if you buy more than one Rank, you care about the cumulative distribution. It's not about rolling a hit on one die, that's always 33.3% -- it's about the chance of rolling cumulative hits across all the dice.

At Rank 1, you have a 33.3% chance of having nullified a hit.

At Rank 2, since you nullify two dice, you have a 55.56% chance of nullifying one hit, and an 11.11% chance of nullifying a second.

Rank 3, you're weighing in at a 70.37% chance of one hit, 25.93% for two, and 3.7% for three.

Rank 4, it's 80.25%, 40.74%, 11.11%, and 1.23%.

It's called a Chi-squared distribution. As you increase your Rank and roll more dice, yes, your chances improve -- but the actual results are modeled not by the odds of rolling a hit on a single die (that is, indeed, always 33.3%) but by the distribution of hits across multiple dice.

As you can see, yeah, Rank 4 gives you a bigger chance of one hit, but it's NOT a 133.2% chance! As I said...twice...Rank 1 gives you 33.3%, but that second Rank only buys you an additional 22.25% for one hit. Because it's about more than one die, now.

Diminishing returns. For reals. Stay away from dice pools and casinos until you grok this.


-Jn-

[Blue Rose]

You're wildly over complicating the statistics to the point that you've lost the forest in the trees.

The probability of negating one hit is not important. The part you care about is how many successes your enemy gets, and that gets abstracted as an average.

Every die you take away reduces the target's average result by one third of a success. That simple. There are no diminishing returns.

And yes, you can have a 133 1/3% chance of negating a success and still sometimes not negate any because you can accomplish the goal of negating a success multiple times on a single roll and every time you accomplish your goal counts equally.

An increase from 33% chance of one to 55% chance of one and 11% chance of two is still a 33% improvement. (55-33)+11=33%. No diminishing returns.

Stay away from casinos until you learn to aim those statistics at the right rules. You can crunch all the numbers in the world, but if you crunch the wrong numbers, you get meaningless rubbish out.

[JoeNapalm]

You're wildly over complicating the statistics to the point that you've lost the forest in the trees.

The probability of negating one hit is not important. The part you care about is how many successes your enemy gets, and that gets abstracted as an average.

Every die you take away reduces the target's average result by one third of a success. That simple. There are no diminishing returns.

And yes, you can have a 133 1/3% chance of negating a success and still sometimes not negate any because you can accomplish the goal of negating a success multiple times on a single roll and every time you accomplish your goal counts equally.

An increase from 33% chance of one to 55% chance of one and 11% chance of two is still a 33% improvement. (55-33)+11=33%. No diminishing returns.

Stay away from casinos until you learn to aim those statistics at the right rules. You can crunch all the numbers in the world, but if you crunch the wrong numbers, you get meaningless rubbish out.

*facepalm*

No. You don't add them.

Those are the odds of rolling AT LEAST that number of successes. That 55.56% chance of at least one hit means a 44.44% chance of ZERO hits. Not 33.3%.

The fact that you're suggesting that you can have a 133.2% chance to roll a hit on four dice shows you have no idea how this works and are not open-minded enough to learn otherwise.

You can easily roll four dice and score zero hits. Nearly a 1-in-5 chance, as a matter of fact. Try it out.

Whether you're rolling four dice or hoping to remove four of their dice doesn't matter -- it's the odds of the number of hits on those dice. In fact, the more dice you opponent is rolling, the less impact this ability has, compounding the diminishing returns.

Don't take my word for it. Look it up. Or flip a coin. By your math, if you flip it twice, one MUST be heads.

Done debating this. Going round and round over ambiguous rules is one thing, but this is cold hard math.

-Jn-

[Kiirnodel]

Cold Hard Math, and you are both right. Because you are talking about different things.

Joe, you are (like Blue Rose said) taking a complex and in depth look at the broad statistics of the dice rolls. Going into the deep aspects of probability and statistics.

Blue Rose, Adamo, etc. are all talking about the simplified raw average chances. They use these numbers because not everyone has studied (or cares to analyze) the complex statistics behind complex probabilities. They are basing their numbers on the raw probability that a single die will get a hit (which is 1 in 3).

If you remove two dice, you are removing two 1:3 chances of getting a hit. Yes, statistics show that this does not remove 2:3, because there are a lot more possibilities out there and complex analysis will show a whole slew of data. But on the fast and narrow, we don't want to add even more math that doesn't directly help us with our goals.

So we simplify, each die is a 1:3 chance of getting a hit. Therefore for every three dice, we will get (on average) 1 hit. Ergo, removing three dice from someone's pool will remove (on average) one hit from their successes. Yes, complex analysis of the statistics and probability shows that removing (or adding) three dice does not guarantee a change of one hit, but the average success rate is all that most people are looking at. We aren't breaking it down into the chances of getting every possible outcome and the inherent probabilities that creates.

[JoeNapalm]

No, no we are not "both right."

If I'm discussing the actual probabilities of dice pools and diminishing returns on Rank investment, and someone comes into my thread and challenges the math based on rolling a single die, they aren't equally correct simply by then saying "I don't care about math."

If you don't care about how it actually works, then don't start %#£¥ on a thread about the math.

This isn't arguing interpretations of RAW vs RAI. This is how dice work. It's not really debatable. You can test it. Grab a handful of dice, start rolling, track your results. We are NOT "both right" and it only takes a few throws to prove it.

Absolutely done here. Dropping the mic. Nothing to be gained from additional ranting. Just throw the dice. If you can fail with three or more dice, you've got your answer.

-Jn-

[Darzil]

If you are discussing the need to get to a certain threshold (eg to hit, to overcome soak, etc), then you need to worry about a lot more than 1/3 of a success, as low rolls fail, but high ones don't improve things.
If you are discussing the average amount of damage something will do, given that it'll be over the threshold, then 1/3 is fine.

[Blue Rose]

Hoo, boy.

*Pops her knuckles.*

Here we go.

Statistics is not math.

Statistics is a logical method that incorporates math as a principal tool.  You can do the math flawlessly, but if you select a nonsensical model or fail to interpret data sensibly, the math becomes useless.

I find your math sound and your statistical analysis to be an abject failure in model design and interpretation.  Good math plus bad logic leads to bad conclusions.

You have correctly calculated the probability that increasing AP values from one to four will prevent at least one success, and assume in your model that preventing two, three, or four successes is a distinction not worth accounting for.

Let us call preventing a success our event.  If your AP manages to prevent a single success, then 100% of an event has occurred.  If you prevent two successes, 200% of an event has occurred!  You can have percentages greater than 100%, depending on your model.  You just have to interpret what that percentage means.  That 133 1/3% does not mean that if you roll three times, there will be four separate rolls in which an event occurs.  It means on average, 400% of an event shall occur overall because an event can occur more than once per roll.

By your own math, of AP -4 having an 80.25% chance of negating at least one success, a  40.74% chance of negating at least two successes, an 11.11% chance of negating at least three successes, and a 1.23% chance of negating four full successes, you agree with me.  Once you count every event, that 80.25% where a first event occurs, plus the 40.74% where a second event occurs, plus the 11.11% where a third event occurs, plus the 1.23% where a fourth event occurs?  That's an average of 133 1/3% of an event per roll!

Here is a big spreadsheet full of lots of random numbers and some math.

What I did here was assume a Body 4 human wearing an armored jacket.  I used randbetween(1,6) to model a 6-sided die, spread across 16 rolls, then columns all the way out to column ZZ.

Below that, I constructed a sum of if(CELL>4,1,0) statements adding up all 16 random die rolls in the respective columns.

Below that, I did the same, counting the top 15 dice, then 14, then 13, then 12 dice.

On the far left, I took the mean of each set of randomly rolled successes for each number of dice rolled.

The results?

16 dice: 5.35 successes on average.
15 dice (AP -1): 5.021429 successes on average.
14 dice (AP -2): 4.698571 successes on average.
13 dice (AP -3): 4.361429 successes on average.
12 dice (AP -4): 4.06 successes on average.

In other words, what I've been saying all along.  Each additional point of AP reduced average number of successes by approximately one third of a success.  Each point of Penetrating Strike was equally valuable, as the goal of the ability is to reduce the average number of successes your opponent scores on soak rolls.  No diminishing returns.

[Adamo1618]

I don't know why you would use a chi-squared distribution, this is a binomial distribution. Number of hits = X~Bin(n, 1/3)

[JoeNapalm]

By your own math, of AP -4 having an 80.25% chance of negating at least one success, a  40.74% chance of negating at least two successes, an 11.11% chance of negating at least three successes, and a 1.23% chance of negating four full successes, you agree with me.  Once you count every event, that 80.25% where a first event occurs, plus the 40.74% where a second event occurs, plus the 11.11% where a third event occurs, plus the 1.23% where a fourth event occurs?  That's an average of 133 1/3% of an event per roll!

That was a well-written post, and I appreciate the civility of the discourse.

I really swore I was walking away from this, but I'll respond in spite of myself. 

You're adding stuff together that doesn't get added together. You can't have a probability greater than 100% in this context. Only the ground has a kill ratio of 1.

What those percentages represent is the probability of rolling at least that many successes. The chance of rolling at least one success includes the probabilities for rolling more than one success. You can't have a 100% chance of rolling a hit on a dice pool, even if you had an infinite number of dice.

Your big table is impressive, but it's not a dice pool, it's a spreadsheet using a simple RNG. You're modeling one die rolled over and over, in sequence (which is always 33.3%) not a big pile of dice rolled all at once (which is vastly more complex).

If your maths and tables were correct, there would be no 3D6 Bell Curve, Gary Gygax would cease to have existed and we never would have had this conversation. 

Here's the real deal:






-Jn-

[Kiirnodel]

You two are talking past each-other, talking about different things.

JoeNapalm is identifying the individual probabilities of several different dice pools and comparing the relative proportions of the chance of getting the various number of hits. This is perfectly identified in the chart he just posted.

Everyone else is talking about the statistically average number of successes one is likely to get based on given dice pools.

Go back into your raw data, using the probabilities of getting exactly a certain number of hits, and take the number of successes that is getting the highest probability for each die pool. More than likely it will be the "average" result, or one third of the number of dice rolled.

Take your chart, the 9 row. If we separate out the data the chances are: 9 hits (0.01), 8 hits (0.09), 7 hits (0.73), 6 hits (3.42), 5 hits (10.24), 4 hits (20.49), 3 hits (27.31), 2 hits (22.41), and only 1 hit (11.71), which leaves a 3.59% chance of getting 0 hits. The result with the highest probability of occurring is 3 hits (at ~27%). As you increase the number of dice the standard deviation increases, making other numbers less "statistically outlandish" but the average or most statically probable result will always be one third of the number of dice rolled.

And no, technically rolling a big pile of dice vs a single die and tabulating the results based on groupings does not have different results. Each individual die is still a randomly generated number between one and six. Rolling six dice simultaneously and a single die six times has the same probabilities of getting any individual set of  six results. There is a bell curve, if you take all of the results on that chart (Rows 18, 20, 22, and 24; Column C - ZZ) and make a Bar graph where the value in the cell is the X axis and the number of times that result occurs is the Y axis you will see your curve.

[Reaver]

<Reads>
<Stares>
<Blinks>
<goes and gets dictonary, chair, and cooler filled with Beer>

<reads more>
<reads more after drinking most of the beer>


Can I get that in "non-uber-math-geek" speak please? Some of us are too drunk to put on our "smrtz hatz:...