Dice vs. Piercing

[DeckBuilder]

STATISTICS has been described as "the science which tells you that if you lie with your head in the oven and your feet in the fridge, on average you'll be comfortably warm"....

That's an example of kurtosis, I believe?
No, it's not, I just checked it.
Go on somebody, what's the polarised "U" distribution (where mean = Normal Distribution) called then?

I am sure that Good Statistics don't lie.
Its reputation is harmed by Bad Statistics that marketers like me use for nefarious "messages"...

Back to the OP:

I'm obviously in the Sane Camp here
You can see that the value balance was appreciated by designers:
Power Strike or Bear Strength vs. Piercing Strike or Critical Strike
Rajan's Fury (situational melee) vs. Tooth and Nail (even ranged)

Are there any situations when +1 Piercing would better (more certain) than +1 Die?
Let's test Hedge's point when excess damage in the distribution curve is irrelevant
Please tolerate cruddy Probability Theory (opening myself to be slaughtered here)
All I remember from my school days is AND = multiply, OR = add, NOT = 1 minus

SCENARIO

I'm attacking something with 1 Armour and 1 Life left with a 3 dice Falcon (0 piercing)
You really don't make any weaker attacks than that in the game really
Would I prefer +1 piercing (Tooth and Nail) or +1 dice (Rajan's Fury)?

With +1 Piercing
Chance of failure = rolling all 3 dice blank = 1/3 x 1/3 x 1/3 = 1/27
Chance of success = 1 - 1/27 = 26/27

With +1 Die
Chance of rolling all 4 dice blank = 1/3 x 1/3 x 1/3 x 1/3 = 1/81
Chance of rolling exactly 1 die with 1 normal damage other 3 blank = 4x (1/6 x 1/3 x 1/3 x 1/3) = 2/81
The "4x" is because any of the 4 dice could be the 1 normal damage
Chance of failure = 1/81 + 2/81 = 1/27
Chance of success = 1 - 1/27 = 26/27

So with 3 dice, in this situation, it seems like +1 Piercing is equal to +1 Die

But what if we attacked this 1 Armour 1 Life object with 6 dice?
Would 6 Dice +1 Piercing or 7 Dice be better?

6 Dice +1 Piercing = 1 - (1/3)^6 = 99.86%
7 Dice = 1 - (1/3)^7 - 7x(1/6x(1/3)^6) = 99.79%

So with more dice (often the case), hitting that Death Threshold is better with +1 Piercing

I've cheated with Excel below (who needs maths when you can simply formula)
When attacking this 1 Armour 1 Life target, what is the probability of killing it?

#   +1 Pierce   +1 Die
1   66.667%   77.778%
2   88.889%   90.741%
3   96.296%   96.296%
4   98.765%   98.560%
5   99.588%   99.451%
6   99.863%   99.794%
7   99.954%   99.924%
8   99.985%   99.972%
9   99.995%   99.990%

So is this the (not-so-subtle) difference between Statistics and Probability?
Statistics deals in averages (but we don't care about the excess damage)
Probability deals in an intended outcome, it's binary, it works or it doesn't

I feel it is linked to Burn = 3 damage fallacy (because infinite series says so)
I always treat Burn as 1.5 damage because that's the median, not the mean
I don't want my Burn damage to be upweighted by the unlikely outlier events

Mathematicians on this forum will no doubt enlighten us Lesser Humans on the error of my thinking here...

[webcatcher]

I always treat burn as zero damage and then I'm pleased if I get something else. In (mage) war you have to be prepared for the worst case scenario.

Sorry for the minor diversion, back on topic now.

[ringkichard]

I always treat burn as zero damage and then I'm pleased if I get something else. In (mage) war you have to be prepared for the worst case scenario.

Sorry for the minor diversion, back on topic now.

This is overly timid, and likely wastes resources and causes you to undervalue agression.  It also doesn't tell you how to evaluate your opponent's burns on your creatures.

[Aylin]

I feel it is linked to Burn = 3 damage fallacy (because infinite series says so)
I always treat Burn as 1.5 damage because that's the median, not the mean
I don't want my Burn damage to be upweighted by the unlikely outlier events

Mathematicians on this forum will no doubt enlighten us Lesser Humans on the error of my thinking here...

If I have time later I'll look over the rest of your post, but from a cursory glance I didn't see any problems.

But estimating the value of Burn at 1.5 assumes it will survive, on average, for just under two turns. (Nitpicking, 1.5 isn't the median of the series either.)

The series in question:
(3/2) * Sum(n=1 to Infinity) (2/3)^n

Expanding:
1 + 2/3 + 4/9 + 8/27 + 16/81 + 32/243 + 64/729 + 128/2187 + 256/6561 + 512/19683 + ...
~= 1 + .667 + .444 + .296 + .198 + .132 + .088 + .059 + .039 + .026 + ...

So the average damage you can expect from your Burn is determined by how many turns you think there is left in the game:
1: 1
2: 1.667
3: 2.111
4: 2.407
5: 2.605
6: 2.737
7: 2.825
8: 2.884
9: 2.923
10: 2.949

When there aren't many turns left in the game, ~2.5 is a good estimation (~3-5 rounds left), but 3 is a good estimation for when there are 7 or more rounds left.

[webcatcher]

@ ringkichard

Not burn on my own creatures, just burns I inflict.  2/3 chance is definitely not good enough for me to count on, so I just think of burn as bonus damage.

[sIKE]

@ ringkichard

Not burn on my own creatures, just burns I inflict.  2/3 chance is definitely not good enough for me to count on, so I just think of burn as bonus damage.

So what math do you use on your own creatures with Burns? Debonus Damage?

[webcatcher]

I don't usually find it that useful to quantify burn tokens on my own creatures. You only roll one die at a time per burn token and at that level the statistics aren't very predictive. For example,  let's say there's a burn token on one of my creatures. On average, it should do three damage.  So in the upkeep phase I roll a 2. What's the expected damage output of that burn token in all future turns? Because of the way the statistics work, it's still three damage, even though it's already done 2. So I just classify burn tokens as "indeterminate amount of future damage" without worrying about how much damage I should expect the token to do.

[sIKE]

I don't usually find it that useful to quantify burn tokens on my own creatures. You only roll one die at a time per burn token and at that level the statistics aren't very predictive. For example,  let's say there's a burn token on one of my creatures. On average, it should do three damage.  So in the upkeep phase I roll a 2. What's the expected damage output of that burn token in all future turns? Because of the way the statistics work, it's still three damage, even though it's already done 2. So I just classify burn tokens as "indeterminate amount of future damage" without worrying about how much damage I should expect the token to do.

Interesting, so how do you determine when/if it is worth to remove them outside of Upkeep?

[webcatcher]

I ask myself two questions.
1. Is this creature's long term survival critical to my plan?
2. If I remove the burn tokens, is the creature likely to survive long enough to be useful?

[DeckBuilder]

+2 Dice > +2 Piercing, always

So with 3 dice, in this situation, it seems like +1 Piercing is equal to +1 Die
...
So with more dice (often the case), hitting that Death Threshold is better with +1 Piercing

from a cursory glance I didn't see any problems.

Thank you, Aylin, but it wasn't me you offended (edited out). As for...

Nitpicking, 1.5 isn't the median of the series either.

I know

However, when you have an infinite series like burn, surely median damage is a more reliable indicator?
In this case 3/2 damage = 5/9 probability.
Therefore median = 3/2 x 9/5 x 1/2 = 27/20 damage = 1.35 median damage.

The reality is, if you test 100 burn counters repeatedly in a Monte Carlo simulation, half the time, they will deal less than 135 damage and the other half of the time, they deal more. With the ability to deal a significant amount in outlier cases. Median identifies its 50th percentile, unbiased by the high damage of those outlier observations which raises the mean.

Now, I don't really understand math (I don't know how long I can keep up this Colombo act) but infinite series burn seems related to Damage Threshold.

For those who want to understand burn better, please reference Zuberi's thread

http://forum.arcanewonders.com/index.php?topic=13139.0

Where Kich made probably my vote for the most laugh-out-loud post on this forum.

This is why, today, the warlock foundation is announcing a very special pledge drive to help at risk burn tokens get through those first critical turns so they can thrive and meet their full mean potential. Just two complete rounds is all it takes for a burn token to do (1.5 * 2) = 3 expected damage. When a warlock comes to your door with her lash of hellfire, we do hope you'll give generously.

[Aylin]

Deckbuilder, I'm not sure where you got 5/9 as the probability for 3/2 damage... in the thread you linked to you list it (correctly) as 2/9. I'm also not sure what exactly you're taking the median of. None of the methods I am aware of for finding a median work for a monotone infinite sequence. What exactly was your thought process?

[lettucemode]

Instead of comparing Dice vs. Piercing in a 1:1 ratio it would be better to compare them at a 2:3 ratio. After all the question is ultimately whether using Power Strike vs. Piercing Strike is better, or whether Bear's Strength vs. Critical Strike is better, right? In both cases you are comparing +2 dice with +3 piercing. Hence a 2:3 ratio.

This is probably the source of the statistical vs. intuitive disconnect that a few people are mentioning. From a gameplay perspective there's no card that gives only +2 piercing.

[Wildhorn]

Piercing +3 will add from 0 to 3 extra damage.
Melee +2 will add from 0 to 4 extra damage.

[jacksmack]

Piercing +3 will add from 0 to 3 extra damage.
Melee +2 will add from 0 to 4 extra damage.

<3

[lettucemode]

Not playing the lottery will earn me between 0 and 0 extra dollars.
Playing the lottery will earn me between 0 and 10 million extra dollars.