Let your intuition work: theoryhammer without maths - P1,2,3

How to beat those cowardly High Elves?

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Let your intuition work: theoryhammer without maths - P1,2,3

Post by Daeron »

Introduction and purpose

Throughout a warhammer game, we often try to guess the outcome of an action, the performance of a unit or spell, to help us build a strategy. More often than not, these outcomes depend on a random factor from dice. But when we're trying to predict dice outcomes, mathematical calculations quickly get difficult (cast value probabilities, for example) or inaccurate (experience shows some units perform better than expected, and some worse).
Some fervently engage in difficult maths and theory to work out itty gritty details of probabilities. But it gets silly when you reason that the "learning experience" for statistics is already there: our own gaming experience, in particular our experience with dice.

I'll try to offer a series of three brief posts, each explaining a way to leverage or sharpen something we already have: our intuition, as it is trained by the game.
With these tricks, I hope people can improve their reasoning and learn to identify how they make better use of their intuitive reasonsing.. or how to better understand a theorycraft without being blinded by seemingly insignificant numbers.

Your intuition is extremely well trained, use it!

The first pointer I'll try to give is "how to read percentages" when reading a theorycraft. If there's one thing a Warhammer player has done a lot, it's rolling a D6. In combat and shooting, we frequently roll them by the dozens! Heck.. If I were to tell you that 83% of the population is literate, you might blink and try to figure out if that number is good or bad. If I were to tell you that 83% equals rolling a 2+ on a D6, then your guts might tell you that's fairly good, but still a risk if it's your only backup from being illiterate.

We can give percentages, we can give averages... but the truth is that such numbers are less likely to feel intuitive as rolling a D6 is. Whenever you read a theorycraft with odds and percentages, try to bring down odds back to a simple thing like a D6 roll, something which you are very familiar with.

Here is a table for reference so they may help you "feel" intuitively what odds are in other posts.

  • 1+ 100%
  • 2+ 83.3%, with reroll: 97.2%
  • 3+ 66.7%, with reroll: 88.9%
  • 4+ 50.0%, with reroll: 75.0%
  • 5+ 33.3%, with reroll: 55.6%
  • 6+ 16.7%, with reroll: 30.6%


Of course, this is assuming we need to roll at least a given score. It's just as easy to roll a 1,2 or 3 than it is to roll a 4,5 or 6. Still, these representations and tests are used so often we can blindly feel how good that number is.

It is possible to expand this table with other kinds of rolls like: "reroll succesful rolls" and other rules we may encounter in the game. However, these aren't used as frequently as the tests above.. and the above reference table should already provide a fairly good start to let your intuition speak.

A brief word to other theorycrafters.

On Druchii.net I see a good effort is spent in breaking down the results of difficult calculations to a usable format. But sometimes.. we fall back on percentages. 90% or 80% doesn't always look so clear to people what that number really "feels like" in the game. Perhaps, we can include such words as "it's like rolling a 2+ on a dice" or "it's like rolling a 3+ with a reroll". It could help people understand what that number means and how it may play out on the table.

Of course, such deductions could be kept to a minimum. I can't imagine seeing a table laying out probabilities become clearer if they all start listen "2+ roll! 3+ roll with reroll!" :D

I believe it's worth keeping in mind though.
Last edited by Daeron on Wed Feb 13, 2013 4:56 pm, edited 2 times in total.
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Let your intuition work: theoryhammer without maths - Part 2

Post by Daeron »

Tests on a few dice rolls: dangerous variation

Rolling to hit, wound, save... we do it so often we almost "feel" our odds without being able to word it in percentages. But we tend to run into trouble when working out the effect on multiple dice. Shades can gun down a warmachine, needing 6's to wound. You could get lucky and kill it in one turn, and sometimes you're gunning it for 3 rounds. We probably had our share of Black Guards needing 3+ to hit with a reroll, then hitting with every attack. While this may not seem like average outcomes, this isn't (or shouldn't be) unexpected. This is variation at work.

It's a fairly intuitive thing. Imagine trying to roll a single 6+ ward save. You're probably going to fail, but there's that chance to get it. If you make it, you made 100% of your saves. If you need to roll two 6+ ward saves, it's quite likely you'll fail both but it becomes quite possible to get 1 save off (or 50%). The odds of getting both (100% of the saves) are really slimming down. As the number increases, getting that very good outcome of 100% becomes increasingly hard. So, as the number of rolls increases, the outcome becomes more stable around the average.

This gives an interesting side effect: the more we invest, the more "average" the outcome will be.. But if that average is quite poor, then it becomes more costly to invest in a poor result. The opposite is also true: we've all seen our heroes fail their hit rolls or only manage half of them when they should have more, but those Black Guards rarely seem to let us down (on the hit rolls anyhow). As the investment (number of rolls) increases, so do the odds of negative outcomes shrink considerably.

Tests on many, many dice rolls! Average is an evil deciever

Another important joy of playing dark elves is the rolling of a vast number of dice. Some of would do a quick calculation, say, Corsairs fighting Dwarven Quarrelers in a second round of combat:
- 36 attacks
- Around 18 hits
- Around 6 wounds
- About 1 save
-> Around 5 kills.

This kind of reasoning will probably do the job if you just want to know if you're going to kill a few Dwarves, but I'm quite certain you'll score less than 5 kills more often than not. With every roll you make, you see that potential kill number go down except for the odd turn of luck. A small unit of Black Guards, fighting T3 and lightly armored troops might look at a similar kill scores on average but often tend to score just that or more. How comes?
Average values may decieve you...

Average values decieve you with possible, but improbable outcomes. Here's a shot at explaining it more intuitively:
Taking our 6+ save, we roll 6 dice. On average, we will succeed 1 save. But we've seen it happen before that a good roll saves 2 or 3. Sure, the odds are slim, but it happens. If you roll all 6 dice a few times, you're probably going to see that roll with 3 saves. Still the average of 1 is correct. If you can score 3 times the average on a single roll, this also means you can score less than average, except the only outcome less than average is 0 saves. So, to be "in balance", for that 1 roll with 3 saves you'll have 3 rolls with 0 saves.. which is 75% of the times we rolled all 6 dice.

As the number of dice increases, this stabilizes a bit, but the odds of getting less than average remain bigger "to compensate for unlikely high outcomes". These high outcomes shouldn't be ignored, they do happen, but be aware that the odds of getting worse than average is actually bigger than the odds of getting better. The average outcome for the Corsairs is "optimistic".

How comes the black guard have the opposite? Ah.. Because the opposite is also true! 10 attacks on a 3+ with a reroll will score 8.9 hits on average. We've seen bad rolls before, where they only hit like 6 or so. But for every low roll.. the average points out there will be an even bigger number of times where 9 or 10 hits will be made. So here, our average is "pessimistic".


And you'll probably have figured out where the middle ground is: where the chance to pass a test/score a kill is exactly 50%. But that one merrits a different part of its own. Still, when calculating outcomes in your head, keep in mind that if your chance to score a kill per attack is less than 50%, your average is somewhat optimistic. This is particularly true when many attacks only yield very small scores.

The great challenge of 50% tests

How I love Witch Elves for this.. And why so many other love or hate them too. In a fight against a T3, WS3 with a 5+ AS, they have a 37% chance to score a kill per attack (like a 5+ roll :)). An executioner has 74% chance. (like a 4+ with a reroll).
If we compare 30 Witch Elf attacks to 15 Executioner attacks, their average is about the same: 11 kills. However... If we would match these odds in my combat calculator here:
http://warhammer.orderoftheathanor.eu/C ... lator.html

Then we'd notice that anything from 8 to 15 kills with be a possible outcome.. A difference of 7 kills! The executioners have anything from 9 to 13. Only a difference of 4. What happens is that, at 50% tests, variation is as big as it can get. This means it literally can go either way... much to your advantage or disadvantage. Try it out on 6 dice. Try to roll 4+ tests and see how often you actually get the average. Try to roll 2+ tests then, and see how often you fail exactly 1.

This is randomness is true in different stages of the rolling. For example, if you need to hit on 4+ and wound on 4+, you can expect a rather random number of hits, and based on the number of hits a rather random number of wounds. In total, though, you won't be scoring that much because you're only looking at 25% chance to score a wound. To put it differently: you need 2 lucky rolls to get lots of wounds in, but only 1 bad roll to get little result.

Things to keep in mind

A lot has been said, but it can be slimmed down to a few simple key principles:
- If your odds of scoring a kill or wound are less than 50%, know that your average outcome is slightly optimistic. If you want a reliable strategy, lower the expectations.
- The closer your odds come to 50%, the more random the outcome will be. Beware of 4+ rolls in hit, wounding or saving. They become random in higher numbers.
- If your odds of scoring a kill or wound are more than 50%, know that the average is slightly pessimistic. This means is a fairly reliable strategy, barring the "utterly bad roll" of course :)
Last edited by Daeron on Thu Feb 07, 2013 10:58 am, edited 1 time in total.
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Post by Daeron »

Let your intuition work: theoryhammer without maths - Part 3


Impact hits, thunderstomps, magic missiles... We have a fair few rolls in our game where the value of the dice roll is used directly to determine the strength of an action. Contradictory to a test, where any set of values (3+, 4+, 5-) are valid, this is an extremely random outcome where average values can fail us.


The value of a single dice roll, there is no "3.5".

The average outcome of a single dice roll is 3.5. But what good does that knowledge do us? 3.5 isn't even a possible outcome. 3 or 4 are no more reliable outcomes than 1 or 6. Every outcome has an equal chance of coming true.
Furthermore, outcome 6 is 6 times as valuable as the outcome of 1. Can we really predict the outcome of an action based on such a dice roll then?

If we assume 3.5 as the outcome of a dice roll, we have a large chance of failing to obtain our expected outcome... and any tactic based on this outcome becomes unreliable.
But if we assume a more pessimistic value, that a chariot has only 2 impact hits (3 if you add scythed wheels) and you reason you still have a 16.6% chance to fail getting that outcome, then chariots start to look very poor on paper! Perhaps worse than they are.

It seems difficult to pin point the exact value of anything that relies on such a roll, but that doesn't mean we can't learn how to cope with that. A simple method is to calculate how much you need, instead of how you expect.
Given the impact of a chariot on a unit, how many impact hits do you need to gain out of the situation? 2+? 3+? By working out how much you need, you gain an intuitive feel of your odds at getting something good out of a move.


We can conclude:
A single dice roll of which the value is directly used for an effect is extremely unreliable. As trivial as it may seem.. keep this in mind. Average values are such a poor guide that it's best to simply stay away from them completely in this situation! Don't base your strategy around the outcome.

Calculate the outcome you need, rather than the average to discover your odds of making it.

The magic that happens when using two dice

As we've shown, the value of a single dice is hard to predict. Contrary to our tests... this improves considerably when working with multiple dice!
When throwing 2D6, the average is expected to be 7. But this also becomes the most likely outcome. Extreme failures, such as double 1, and extreme successes such as double 6 become drastically smaller: less than 3% (like a 2+ rerollable save).

To my knowledge, there is no test, no roll where the behavior of odds change as dramatically as they do when moving from D6 to 2D6. In spite of the fact that 2D6 still has its share of randomness, the reliability of 2D6 is miles ahead of 1D6. And while rolls like 3D6 improve the odds of getting an average outcome, they do not increase the stability by the same factor.

This is important to keep in mind, in particular when using chariots. I believe this is the very reason why 2 chariots are more than the sum of their individual performance. This is why a 2D6 magic missile is significantly stronger than a D6 magic missile, but a 3D6 magic missile may not be worth its investment compared to its 2D6 variant.

The actual odds when using two dice are, unfortunately, more difficult to calculate... but here we can summarize some theorycrafting to help us out:
- 5+ on 2D6 has 83.3% chance, which is like rolling a 2+ on 1D6. It has a risk, but a fair shot of succeeding.
- 7+ on 2D6 has 58.3% chance, which is like rolling a 5+ with a reroll on 1D6. I wouldn't bet my army on it, but it has a good chance.
- 10+ on 2D6 has 16.6% chance, which is like rolling a 6+ on 1D6. Hardly reliable.. but it can happen. Use it only if the gain is worth the investment and risk.


Going beyond two dice

As a brief discussion point, it's interesting to examine what happens when rolling 3D6 or more. Without special rules (like irresistable force for spells), rolling more than 2 dice will steadily augment the outcome with each dice, but here's an interesting fact:
- The chance of getting 8+ on 3D6 is nigh the same as 5+ on 2D6
- The chance of getting 11+ on 3D6 is nigh the same as 5+ on 2D6
- The chance of getting 14+ on 4D6 is nigh the same as 5+ on 2D6.

While the values steadily increase, and become quite attractive if you're looking at hits, you can also see that you're only getting 3 reliable hits for every dice... even though the average is 3.5.
What happens is simply that the number of outcomes increases significantly with every dice and unfortunately this translates to more spread out results, and thus less reliable high scores. On the bright side, the potential output does increase, and increases significantly. This is why, say, charging a unit with 3 chariots is certainly still worth while. This is why the third rank of fireball costs so much more and may still prove worth it.
But you've probably reached the optimal investment at 2D6. It's the smallest number of dice where the odds of getting minimal outcomes decrease significantly, while the potential increases notably.
Last edited by Daeron on Wed Feb 13, 2013 4:54 pm, edited 1 time in total.
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Post by L1qw1d »

for some reason, i'm reminded of Settlers of Catan's distribution array of how probable rolls are lol

We don't have to handle standard deviation, right?
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Post by Daeron »

Added the second part.

We won't dig as deep in theory though variation is mentioned. The point isn't to get more theory in, but simply to keep a few pointers in mind: when is our intuitive reasoning correct, when do we need to sharpen it a bit...
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Post by Trax »

This is a really interesting and quite important reminder, especially for someone like me who tends to only keep the averages in mind and forgoes any deviations. Eagerly awaiting the third part.
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Post by Red... »

Excellent article. Roll eighteen dice and you should get three sixes - but you may get eight, you may get none. Roll two dice and you should get a seven, but you may get a two and you may get a twelve. I always think most about the old Assassin with RoK and ToD versus Chaos Lord thing - people used to come on these forums complaining that 'I hit him seven times, so I should have managed at least one killing blow, but I didn't.' Should have, smud have.

That said, the easiest way to get around fickle dice not managing averages is to stack the deck so heavily in your favour that you CANNOT within fundamental reason fail to pull off what you need to do. 41 WS5 S6 hatred attacks from a COB blessed horde of Execs WILL (almost) guarantably slay a unit of 8 knights with a 1+ or 2+ save, because even if you roll an absurd number of 1s and he makes an absurd number of saves, you will still wipe him out to a man.
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Post by Omnichron »

It's a good thing to have in mind... percentage chances are kinda so and so. In a game you might think that "LD 10 is good enough for CoK, so I'll make sure my BSB is covering another part of the army where I might need it", but suddenly you roll a bad LD and end up in a bad position or not getting to use the unit as you needed.

The thing that have made me improve from just a mediocre player, is how I have changed the whole idea about "This should happen" aspect. Earlier I'd think a lot about how I'd do if I managed to roll average, and if the average would do, I might go for it. I would take chances like moving my sorceress unit within the range of a gateway distance, as it is a really slim chance of him getting it through my magic defences AND getting 11+ roll on the strength dice... but it happens. The last tournament I was in, I did lose my sorceress that way, although the reason it happened was that I was sloppy with my measuring. My tactical decision about the matter was to stay just outside the range of the spell altogether (But I failed the simple task of measuring correctly)...

And that is the main change of my strategies now. A 5% chance of failing is a 95% chance of success... however if you can get a 100% chance of success you should ALWAYS go for that.

Of course, you can't base your strategies on getting all those 100% chances, but you should limit the % chances of failure as much as you can... It's also funny that the witch elves are mentioned, because they are so random, and yet I use them in 95% of my lists (and they have been the best unit in all the tournaments I've had them). Why do they perform so well?!

Because I limit the percentages of failure by:
1. Making sure I don't rely on them to win my games (So, smaller units which I can sacrifice).
2. Debuff my opponent with Shadow (Miasma, Withering, Enfeeblement) or boost the witch elves with mindrazor.
3. Use CoB to give them extra attacks so that rolls below average still do a lot of damage.
4. Have the Banner of Murder so they can take down medium armored foes.

The last time I lost them by taking some chances was when I went in to kill chaos knights. I needed to get through mindrazor to take the knights down, but got 3 PD that magic phase. Why did I do that risky move? Because I could handle the rest of the army and was ready to sacrifice the witch elves. The battle ended with me winning 20 - 0 even while my witch elves got slaughtered... if the mindrazor had got through, the killing of the knight bus would have been a bonus.
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Post by L1qw1d »

Lol Also sad there's not a 'and people have a tendency to flop dice when they need good rolls' deviation ;)
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Post by Red... »

Nowadays I always play with the assumption that:
- My dice work for the enemy.
- After rolling a ton of 1s and 2s for my to-hit and to-wound rolls, the bell curve will swing around and slap me in the face with lots of 5s and 6s for my leadership tests.
- The words 'all I need to do is roll anything other than a double one' will automatically produce a snake eyes result.
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Post by Omnichron »

Red... wrote:Nowadays I always play with the assumption that:
- My dice work for the enemy.
- After rolling a ton of 1s and 2s for my to-hit and to-wound rolls, the bell curve will swing around and slap me in the face with lots of 5s and 6s for my leadership tests.
- The words 'all I need to do is roll anything other than a double one' will automatically produce a snake eyes result.


Yeah, I have the same assumptions... like when using the sac dagger and already got one 6 in the table, I'll NEVER do that extra safcrifice... because what's going to happen is that I get another 6, then roll 2-4 on misscast, and fail the 4+ test. It happened 3 times in a row with different opponents the last times I did such sacrifices.

I also have the record of high LD rolls on my club as far as I know, failing about 75% of my LD 8-10 rolls during a battle against Wood Elves. I've actually started to count my LD fails and figured out I have a fail rate of about 50%, even with LD 9.

Luckily, I have about the same rates to hit and sometimes to wound... hmm, I wonder if the blue GW dices have some weights to them.
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Post by Red... »

Slight tangent, but here's a fun (very old) thread from warseer:

http://www.warseer.com/forums/showthrea ... -must-read
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Post by Daeron »

Hehe. Yeah, you could go for real dice... But if the article is correct that the inaccuracy is caused by flaws inside the dice, then... doesn't that also mean it's possible for a dice to roll more 6's?
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Post by Killerk »

but if they roll less "1's" then in result your roll's will be better.
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Post by Dyvim tvar »

Looking back at the first post, one of the things that should be apparent to everyone is the power of re-rolls. They greatly increase your chance of success, nearly doubling it when the chance on a single die is low. It can be really nice to be a Dark Elf with that innate Hatred sometimes.

Omnichron wrote:Yeah, I have the same assumptions... like when using the sac dagger and already got one 6 in the table, I'll NEVER do that extra safcrifice... because what's going to happen is that I get another 6, then roll 2-4 on misscast, and fail the 4+ test. It happened 3 times in a row with different opponents the last times I did such sacrifices.


That's the beauty of the Sac Dagger -- it helps you avoid miscasts by rolling fewer dice. If there is already a 6 on the table, you can then decide whether an extra die is worth it or not.

If I have already met the casting value, I normally leave things alone and stop there.

If it's the end of the magic phase and I haven't met the casting value but it's not a critical spell, I will normally just stop rather than risking even the small chance of being sucked into the warp or having my levels drained.

If it is the beginning of the magic phase and a failed casting will effectively stop multiple spells, then I will generally take the risk and roll the extra die.

Or if it is a spell that I really need at a critical junction of the game--my frenzied corsairs are engaged with Mournfangs and I'm trying to cast Mindrazor--I'll definitely roll that extra die (probably hoping for another 6).
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Post by L1qw1d »

I think it's interesting some of the logic that gets missed too. Every die is a 1 in 6 chance to hit any ONE number. More dice just means more 1 in 6 chances. the Better a skill, obviously the better the chances, but that's digression.

The other option in logic that people take is that we need to add all the probable dice all up. That kinda only happens during the magic phase when we're trying to beat that set number.

Other than that we're all just hoping for a bunch of 6's or 1's (which when I hear of 'lucky rolls' given statistics? I think someone rolled a 7 on their 6 sided die)
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Post by Red... »

All true, but math hammer has it's place. Complex stats stuff aside, roll twelve dice and you are likely to get around six rolls of four or above. You are unlikely to get six sixes.
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Post by L1qw1d »

Well, that's what I mean- I wasn't trying to math hammer. I was agreeing with the main post. It's a logic we have to cultivate to teach our minds to say 'we'll get under half' in most pool rolling.

Even if we're rolling one die, it's going to have at least ONE chance of fail. Dunno, I've had to look at statistics a lot lately, and see how people play with them to get what they want- but keeping your logic and wits about you, you get to do what Daeron is noting. Dice don't work FOR or AGAINST anyone. They just roll. People even develop 'methods, habits and superstitions' about how dice GET rolled.

I take Daeron's thread as excellent because it's showing the real world interaction between the logic, the math, and how it all shakes down. It's an excellent synthesis :)
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Post by Trax »

Daeron wrote:Hehe. Yeah, you could go for real dice... But if the article is correct that the inaccuracy is caused by flaws inside the dice, then... doesn't that also mean it's possible for a dice to roll more 6's?


Quoting directly from Red's link:

Game room logic, poor source of anything, would dictate that the side with the one is heavier and would therefore be on the bottom more. Unfortunately this is just not true, take popcorn or batholiths as an example. The 6 is too light to stop the momentum of the dice, the rounded corners cannot prevent the dice from turning due to the weight. In the end 1s are by far the most common result. On a 6 sided dice any given number should appear 16.6% of the time, the Vegas dice were dead on and the square dice with pips were pretty close only displaying a 19% ratio for ones.


This is just an assumption but with their sample size it's pretty safe to say that it's not the case that "some dice might roll more 1's, some might roll more 6's" but rather that the curved rounds and general weight distribution of a common gaming table die is way off - we're talking almost double the odds the respective event should have, almost 30% is huge.

Don't even know what to think of this. I'd rather not have known this fact in the first place, now I can't see myself stop worrying about it...
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Post by L1qw1d »

Don't worry about that at all Trax :lol:

Scupting, model/ statue making and mould casting have been around for oh say...most of civilisation. Trust me, they have the process down pat ;) Getting a few squares shaken for the bubbles to come out is actually IN the design specs for reputable companies.

Worry more about people that just drop dice on the table or throw like it's craps to make more bounces lol
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Post by Daeron »

And part 3 is up. Thank you all for reading this so far :)
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Post by Calisson »

You could have added on part 3 what becomes with rerollable dice.
Is a rerolled D6 (hatred, the OTS) as reliable as 2D6?
Is best 2 of 3D6 (cavalry charges) as reliable as 3D6?
What about limited rerolls (Heavens Lore rerolling 1s or 6s)?

In order to get a good grasp on rerolls, play Blood Bowl! :D
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Post by Lorddrittz »

Really nice article Daeron.

It shows how powerful our Hatred is and the BG re-rolling every round. Like wise the ASF Banner giving re-rolls.
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Post by Daeron »

Calisson wrote:You could have added on part 3 what becomes with rerollable dice.
Is a rerolled D6 (hatred, the OTS) as reliable as 2D6?
Is best 2 of 3D6 (cavalry charges) as reliable as 3D6?
What about limited rerolls (Heavens Lore rerolling 1s or 6s)?


Well.. I'm not sure I see the value of comparing these rolls simply because I don't know a situation where you get to choose between them. The choice between 1D6 and 2D6 may be more common when using ranks of spells (searing doom, fireball, scattering of templates). I gave the example of the chariots to support Red's D.R.A.I.C.H. on chariots, and his theory that it's better to use 2 chariots (and 3 to mitigate risks) than just 1, and part of the reason why that is so.

I just wanted to give a few concepts to people who use calculations in their game to know whether or not they do the right thing by shooting that unit or not, or by charging that unit or another. Dark Elves often have the initiative in these things, and thus it's not uncommon.

Detailed theorycrafts can help to measure the impact of something specific, but I think it's best to do that prior a specific matchup or afterwards to increase the learning experience. "Hey, he did this spell on me.. and a lot of things went wrong since. Is that always going to have such a big impact? Let's calculate!".
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