Paladins vs Terminators Math and Stuff

[Plasmablasts]

Just regarding the relative resilience of Paladins and Terminators, I've noted that a couple of times, Seizeman has asserted that Paladins are 100% more resilient against Dd3, Dd6 and D2 weapons. This is entirely true for D2 weapons, but not for the others I believe. I suspect Seizeman has made an erroneous simplification by treating Dd3 weapons as on average D2. The maths for working out the number of unsaved wounding hits to kill a multi-wound model becomes less straightforward if using variable damage weapons (and I'm happy to be corrected if I've got it wrong!). By my calculations, sixteen unsaved wounding hits from a Dd3 weapon would kill nine Paladins or 12 Terminators. Therefore, Paladins are 33% more resilient against Dd3. Against Dd6, Paladins are 16.7% more resilient.

 

I'd be happy to show my workings if needed.

[Gentlemanloser]

16 unsaved d3 hits would kill 8 paladin.

 

Ave dam for d3 is 2. It takes two unsaved hits to on average kill a 3W Paladin.

 

The 4th point of damage is wasted.

 

Edit.

 

The same unsaved wounding hits would kill 16 terminators.

 

As each hit deals 2W and kills a single Terminator.

 

Edit. The average damage for a d3 weapon is (1+2+3)÷3 = 2

Edited August 29, 2017 by Gentlemanloser

[Seizeman]

The actual average of a 1d3 weapon against a 3 wound target Is 1.56 (if you simplify on the average of 2 dmg, is 2 wound to kiil it, so 1.5 average, which is pretty similar). A 3 dmg roll has the same chance to do 1, 2 or 3 wounds, a 2 dmg roll has 2/3 of doing 2 dmg and 1/3 of doing 1 dmg, a 1 dmg roll always does 1 dmg. The average dmg on a 1d3 damage roll vs 2 wounds models is 1.67. So it takes 1.2 d3 dmg wounds to kill a terminator and 1.92 d3 dmg wounds to kill a paladin. That's a 60% increase in survivability.

 

On 2 dmg weapons, it is much more simple. You need 1 hit to kill a terminator, 2 to kill a paladin, so 100% increase in survivability.

 

On 1d6 damage weapons, the terminator has a 17% chance of surviving, while the paladin has a 33% chance of surviving. That's a 100% increase.

 

That's why when I argue about Paladins survivability, I say they are at least 50% more survivable and up to 100% against some weapons, but if they were just 50% more survivable against everything they would be worth the cost increase anyway. Of course those chances can be altered slightly depending on the exact order in which the opponent shoots his weapons and the use of rerolls, but never to the point where they are less efficient than terminators.

 

The only exception are 3 dmg weapons, which kill them just the same. Luckily, those weapons are rare and found mostly  on dreadnoughts, which is one of the reasons I've advocated for the use of the stormraven, as they can easily kill a dreadnought in a single turn, eliminating the problem before you alpha strike. Anyway, that's not that important now that the codex has been released. Now we don't need to use only paladins as our tanking units, as we now also have the GMNDKS as competitive options for the same role (regular NDKS are not threatening enough for this role, as I've explained some times before). Not only are GMNDKS a lot less vulnerable to dreadnoughts, but they can also kill them with relative ease in the shooting phase, specially with the psychic onslaught stratagem (a GMDNK with onlslaught averages 7 wounds against a dread). The existence of vortex of doom also helps in this situation.

Edited August 29, 2017 by Seizeman

[Plasmablasts]

16 unsaved d3 hits would kill 8 paladin.

 

Ave dam for d3 is 2. It takes two unsaved hits to on average kill a 3W Paladin.

 

The 4th point of damage is wasted.

 

Edit.

 

The same unsaved wounding hits would kill 16 terminators.

 

As each hit deals 2W and kills a single Terminator.

 

Edit. The average damage for a d3 weapon is (1+2+3)÷3 = 2

In 1/3 cases, one unsaved wounding hit causes three wounds, killing a Paladin.

 

In 1/9 cases, it takes three unsaved wounding hits to kill a Paladin (I.e. rolling two single wounds consecutively).

 

On the remaining cases (5/9), it takes two unsaved wounding hits.

 

So, for nine Paladins, one will be killed by three unsaved wounding hits, five by two and three by one: total of sixteen unsaved wounding hits.

 

For Terminators, in one out of three cases, you'll roll one wound and therefore need a second unsaved wounding hit to kill. In the other two cases, you roll two or three wounds, either of which will kill a Terminator in one go. So, for three Terminators, you will on average need four unsaved wounding hits to kill them all (which is the same as needing sixteen unsaved wounding hits to kill twelve Terminators).

Edited August 29, 2017 by Plasmablasts

[Gentlemanloser]

All the damage results are independant of each other though.

 

In 16 unsaved wounding hits each individual hit will on average deal 2 damage.

 

Your working from the chance that in 16 shots 10 are a result of 2, 3 is a result of 3 and 3 are a result of 1.

 

Each result is equally as possible as the other, at a 1 in 3 chance.

 

Out of 16 unsaved wounding hits its just a possible to have 16 results of 3, or 16 of 1.

Edited August 29, 2017 by Gentlemanloser

[Seizeman]

You are assuming that a 3 dmg roll always does 3 dmg and kills a full paladin, which is not correct. A 3 dmg roll agains a paladin with 2 wounds left does 2 wounds, so it kills 2/3 of a paladin. A 3 dmg roll agains a paladin with 1 wound left does 1 wound, so it kills 1/3 paladins. A 3 dmg roll does 2 wounds on average,  a 2 dmg roll does 1.67 wounds on average, and a 1 dmg roll always does 1 damage. As the 3 results have the same chance, each failed save does 1.56 wounds on average.

Edited August 29, 2017 by Seizeman

[Gentlemanloser]

A 3 damage roll always deals 3 damage.

 

Regardless of how many wounds the paladin has left.

 

The excess is wasted, but that's not the point.

 

Otherwise we might just as well claim a paladin has on average 2 effective wounds.

 

As they could be on 1, 2 or 3 remaining wounds, and on average be killed outright by 2 damage unsaved wounding hits.

 

Edit. And that would only apply to the first mini in a unit, as they might have previous damage. But all other paladin in the unit are in full wounds, all the time, due to wound allocation rules.

Edited August 29, 2017 by Gentlemanloser

[Seizeman]

How is that not the point? That's precisely the whole point of the exercise.

 

Who cares about how much overkill some weapon does to his units? The relevant information is how much wounds they take in reality so you know how many models you are loosing, so taking into account excess wounds is necessary to calculate to calculate a true average.

 

What you are saying is that a tactical marine unit taking 6 damage from a single lascannon is the same as they taking 6 damage from an assault cannon, which makes no sense at all. How many models actually die is what matters.

[Gentlemanloser]

Not at all.

 

Your changing the damage value of the weapon.

 

We were only looking at d3 weapons for the last couple of posts.

 

Obviously the numbers change when you change weapon damage.

 

Paladin have no degredation.

 

A 3 damage hit versus a paladin with 1w left doesn't kill 1/3 of a paladin.

 

It kills a paladin outright.

 

Sure same as a 1 or 2 damage hit would.

 

But we're working with 3w paladin here, not partially wounded squads.

Edited August 29, 2017 by Gentlemanloser

[Plasmablasts]

Sorry, wasn't clear enough. My assumptions are based on previously unwounded models: therefore, for example, one in three previously unwounded Paladins will be killed outright by an unsaved hit from a Dd3 weapon.

 

To kill an unwounded Paladin with two unsaved hits, the damage rolls would be either 1 followed by 2 or 3 (probability being 1/3 * 2/3 = 2/9) or 2 followed by any result (probability = 1/3 = 3/9). So, probability of a kill with two unsaved hits is 5/9.

 

Finally, the damage rolls to result in a kill from three unsaved hits are 1 then 1 then anything: probability = 1/3 * 1/3 * 1 = 1/9.

 

Hence, average number of unsaved hits to kill a Paladin = 1/3 *3 + 5/9 * 2 + 3/9 * 1 = 16/9.

[Valerian]

Averages require a data set, which means you look at dice already rolled and use the average as a way to describe central tendency.  What you should be doing instead is describe the probability of certain outcomes, which is what you want to know before any dice are rolled.

 

So, for the record, a d3 doesn't have an average until you've rolled it a bunch of times and examined the results.  Those results are going to vary (most likely) every time you take a set of rolls and examine the new data.

 

V

[Gentlemanloser]

Sorry val, should we have used mean? ;)

 

Plasma, shouldn't it be that there are 7 discrete outcomes;

 

1. 1+1+1

2. 1+2

3. 1+3

4. 2+1

5. 2+2

6. 2+3

7. 3

 

When broken down into the number of hits required that's;

 

1. 3

2. 2

3. 2

4. 2

5. 2

6. 2

7. 1

 

Or 14/7 with a mean of 2?

 

So on average ( using mean for average) it take two unsaved wounding hits to kill a single full wound Paladin?

Edited August 29, 2017 by Gentlemanloser

[Valerian]

Out of 16 unsaved wounding hits its just a possible to have 16 results of 3, or 16 of 1.

And this here is exactly why you can't talk about averages until after the dice have been rolled.

[Plasmablasts]

Sorry val, should we have used mean? ;)

 

Plasma, shouldn't it be that there are 7 discrete outcomes;

 

1. 1+1+1

2. 1+2

3. 1+3

4. 2+1

5. 2+2

6. 2+3

7. 3

 

When broken down into the number of hits required that's;

 

1. 3

2. 2

3. 2

4. 2

5. 2

6. 2

7. 1

 

Or 14/7 with a mean of 2?

 

So on average ( using mean for average) it take two unsaved wounding hits to kill a single full wound Paladin?

Don't think that works: it would mean you had a 1/7 chance of rolling a 3.

[Seizeman]

What's exactly the point of talking averages after the dice have been rolled?

 

The average asumes the first paladin to receive a wound has the same chance to have to have 1, 2 or 3 wounds remaining. Of course that's not the case and it's a bit more likely for the first paladin wounded to have 3 wounds remaining, but the exact rate is impossible to calculate and the difference in the final result is ridiculously small, so it's not worth accounting for it.

[Gentlemanloser]

Only a 1 in 7 chance to only require 1 unsaved wounding hits.

 

Which on a single die roll would happen 1 in 3 times.

 

Heh time for a rest. ;)

[Plasmablasts]

Only a 1 in 7 chance to only require 1 unsaved wounding hits.

 

Which on a single die roll would happen 1 in 3 times.

 

Heh time for a rest. ;)

:) I find walking the dogs the best time to contemplate mathammer!

[Valerian]

What's exactly the point of talking averages after the dice have been rolled?

 

The point is that is what averages are.  There is no such thing as an average of data that does not exist, before any dice have been rolled.  

[Beams]

Why not run a simple simulation? It would be incredibly easy to program and would answer these questions much easier and with less debate.

[Plasmablasts]

 

 

What's exactly the point of talking averages after the dice have been rolled?

The point is that is what averages are. There is no such thing as an average of data that does not exist, before any dice have been rolled.

True, but one can predict averages based on the mathematics of probability.

[Seizeman]

 

What's exactly the point of talking averages after the dice have been rolled?

 

The point is that is what averages are.  There is no such thing as an average of data that does not exist, before any dice have been rolled.  

 

You are mixing up rational data with empyrical data, they are both information, just of a different kind. Don't forget that the point is to determine how efficient a unit is based on their stats, and not to predict their exact behaviour in an actual case.

 

Why would we use a simulation? A simple average is more precise and easy to do

 

Anyway, the point was to show if paladins are more efficient, durability wise, than terminators, and they are in any case.

[Valerian]

Why not run a simple simulation? It would be incredibly easy to program and would answer these questions much easier and with less debate.

 

That's just what I had been working on.  Using excel, I did 5 different trial runs of 1,000 d3 dice rolls, and then looked at the outcomes.  

 

For simulation #1, a 1 was rolled 332 times, a 2 was rolled 329 times, and a 3 was rolled was rolled 339 times.  Thus, in this simulation, a Terminator would have been killed 668 times, while a Paladin would have been killed only 339 times.  This is why Siezeman has stated that Paladins are 100% more survivable against d3 weapons, for example, as they would have been killed outright roughly half as often.

 

The other four simulations resulted in pretty much the same outcome.  

 

Doing a similar simulation using a d6 for 1,000 rolls resulted in a Terminator dead from a single blow 858 times, whereas a Paladin dies 694 times.  

[Valerian]

 

 

What's exactly the point of talking averages after the dice have been rolled?

 

The point is that is what averages are.  There is no such thing as an average of data that does not exist, before any dice have been rolled.  

 

You are mixing up rational data 

 

What exactly is rational data?  I can't seem to find a definition for that term.  The point that I've been trying to make here is that words mean things, and folks are using the term 'average' in a way that is not accurate.  There is no such thing as an average of hypothetical/theoretical data. 

[duz_]

How do you account for damage being wasted? Because D3 is random

 

Eg Damage results / rolls : 1, 3, 2, 1, 2, 3

 

In the above case 4 terminators would die and 3 paladins would die, but you would lose 4 and 3 excess damage respectively... 

If you didn't account for that you would simply say 6 terminators died and 4 paladins.

[Plasmablasts]

 

 

 

 

What's exactly the point of talking averages after the dice have been rolled?

The point is that is what averages are. There is no such thing as an average of data that does not exist, before any dice have been rolled.

You are mixing up rational data with empyrical data, they are both information, just of a different kind. Don't forget that the point is to determine how efficient a unit is based on their stats, and not to predict their exact behaviour in an actual case.

 

Why would we use a simulation? A simple average is more precise and easy to do

 

Anyway, the point was to show if paladins are more efficient, durability wise, than terminators, and they are in any case.

A properly constructed simulation might be worth running if there is doubt which probability model is correct, as a hypothesis test.

 

There's no doubt that Paladins are more resilient: the question was more: how points-efficient is that gain in resilience, which requires an accurate prediction of their durability. A 100% gain in resilience against D2 weapons is unquestioned and that's clearly an important bonus, given the prevalence of super-charged plasma (plus autocannons). In fact, that might be arguably enough to secure Seizeman's argument, regardless of the true effects of Dd3 and Dd6 weaponry. Still, I didn't want to leave what I still think is an incorrect statement unchallenged - and I think the maths is quite fun too!