twity
Junior Strategist
Posts: 179
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Post by twity on Sept 28, 2017 1:18:38 GMT
Let me start this by saying math is not my strong suit, so please correct me if I say something wrong.
I was wondering if we could start a discussion on common dice math misconceptions. I rely on dice math pretty heavily in game, and I hear a lot of things that don't seem to make sense. For instance, if you are damaging something at dice -5, logic would suggest that you should average two damage per model that hits given an average dice roll of 7 and each higher and lower roll having a reciprocal dice roll at equal probability. But this discounts that everything 5 or below is treated equally, so a true average doesn't seem to work. In my mind you would need to take the average of all dice that are 6 or greater, average those, and then multiply that number times the number of times that you would roll over a 5.
So if the average damage at dice -5 of all dice over a 5 are equal to 3.15 (according to my math), and you will roll over a 5 on 26 of 36 rolls, then 3.15 * (26/36) = 2.28. I know this isn't a huge difference, but logically the effect is amplified at every step. So at dice -6 you should average 1.55 instead of 1. This kinda thing can add up. If you attack with 10 infantry, you have roughly a 25% chance of rolling boxcars ((35/36)^10).
What are some other math mistakes you see people make frequently in a game? What am I assuming in a game that is probably off?
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Post by Trollock on Sept 28, 2017 5:58:26 GMT
The most common misconception that "bad" players make is that "you reliably hit if you need to roll a 7". you have a ~58% chance of hitting if you need to roll a 7, so that means if you roll 10 attacks, you should hit about 6 of them. If you roll 5 attacks, you should hit about 3 of them. I often see ppl being disappointed that a jack/beast missed two of their attacks when they needed to roll 7s to hit.
Another common misconception i see all the time is that adding an extra die to a damage roll is some how more effective if you are already rolling lots of dice. For example if a caster has battle lust (additional dice to melee damage rolls), lots and LOTS of ppl seem to think that it is some how better to put battle lust on a unit with weapon master, than one without, when what they SHOULD be looking at is putting it on a model/unit with multiple attacks.
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Post by celeb on Sept 28, 2017 6:54:17 GMT
As a follow up, an unboosted 8 is around 40% hit chance, while some people think that this should be impossible to hit. If you have rerolls or just enough attacks, an unboosted 8 isn't that uncommon.
Sometimes, people debate spending a focus point on a caster when damage comes in that does 5 or 6 points, to reduce damage of higher rolls. While this is important in hordes, 5 damage reduced is 5 damage reduced, no matter how high the damage roll is.
Also, people often underestimate what a +1, +2, -1, -2 can do to the dice math. A +2 on Damage is already huge, as is a +1 to attack and damage rolls.
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unded
Junior Strategist
Posts: 760
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Post by unded on Sept 28, 2017 7:02:01 GMT
So you've hit on a common misconception twity, but your solution isn't accurate.
In order to get the expectation, you need to do away with the notion of an "average" roll. It's not a big loss, as the term "average" is used very loosely in layman-speak, used interchangeably for very specific (and different) statistical measures like "mean", "median" and "expectation". For the record, the 7 on 2d6 is a median value.
When working out your expected damage, the quantity you want is the "expectation". You calculate this by summing the probabilities of each value multiplied by the reward of each value. If both the probability distribution and the reward distribution are symmetrical about the median value, then and only then can you use the shortcut of just assuming the median value. The probability distribution of 2d6 is symmetrical, but the 2d6-X is often not symmetrical. A good illustration of this is looking at 2d6-7, like say a mage-hunter strike force unit shooting into Feora2. if we simply assumed the Median, we would expect zero damage from the unit and then try to hand-wave away the inevitable damage as "spikes". The actual expectation of 2d6-7 is 35/36 (so approximately 1) damage per shot, so a unit of 10 under Ravyn's feat can be expected to put out 10 damage to Feora (obviously not accounting for spending FOCUS to negate damage). Calculating the actual value at the tabletop is pretty tricky, but you can use a rule of thumb for fudging the answer to close enough for gaming purposes. The larger the negative, the higher the level of asymmetry and thus the more you can add on to the median value expected roll. For example while dice-3 has some asymmetry, the only pairing that is asymmetrical is [2] vs [12], both of which have a low (1/36) probability of occurring, and can be reasonably ignored. For reference, you can assume the following values for 2d6 as a shorthand for calculating expectation on the tabletop:
2d6-9: 9.3 2d6-8: 8.5 2d6-7: 8 2d6-6: 7.5 2d6-5: 7.2
Above and below those bounds you might as well ignore the effect. At 2d6-10 or worse, while the assumed 2d6 value seems high the expected damage outcomes are low enough that you simply don't care. At 2d6-4 and better, you are getting close enough to 7 that the error made is small enough that you just don't care.
Now, if you really want to taunt me into a huge wall of text, ask me about what "lucky" really means, and we can discuss Variance.
-und_ed
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Post by Guessed on Sept 28, 2017 11:08:42 GMT
For the record, the 7 on 2d6 is a median value. 7 on a 2d6 roll with no other qualifiers is mean, median and mode all at once (and the expected value too). SInce these three measures are the most commonly used forms of "average", it is entirely correct to say the average roll on 2d6 is 7. Just enjoying the chance to point this out. Carry on.
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whydak
Junior Strategist
Posts: 288
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Post by whydak on Sept 28, 2017 11:31:29 GMT
For reference, you can assume the following values for 2d6 as a shorthand for calculating expectation on the tabletop: 2d6-9: 9.3 2d6-8: 8.5 2d6-7: 8 2d6-6: 7.5 2d6-5: 7.2 Above and below those bounds you might as well ignore the effect. At 2d6-10 or worse, while the assumed 2d6 value seems high the expected damage outcomes are low enough that you simply don't care. At 2d6-4 and better, you are getting close enough to 7 that the error made is small enough that you just don't care. Now, if you really want to taunt me into a huge wall of text, ask me about what "lucky" really means, and we can discuss Variance. -und_ed I would like to read this wall of text I think I have general idea how this works but never seen WM&H specific numbers for this.
Comming back to: 2d6-9: 9.3 Last number is expected dmg? Example is fine but I don't understand this list for different 2d6-X.
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Post by smoothcriminal on Sept 28, 2017 11:56:46 GMT
So does anyone know if hand of fate effect (roll 1 more, drop the lowest) works the better the more dice you roll? i.e. it statistically improves average roll more on 3d6 than 2d6. I felt for a long time that it does, but never bothered to find the math for it.
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zich
Junior Strategist
Posts: 690
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Post by zich on Sept 28, 2017 12:13:09 GMT
smoothcriminal It does. The value of "add one die, drop the lowest" is ~1.5 for 2d6, ~1.7 for 3d6 and approaches 2 if you add even more dice.
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unded
Junior Strategist
Posts: 760
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Post by unded on Sept 28, 2017 12:27:24 GMT
For reference, you can assume the following values for 2d6 as a shorthand for calculating expectation on the tabletop: 2d6-9: 9.3 2d6-8: 8.5 2d6-7: 8 2d6-6: 7.5 2d6-5: 7.2 Above and below those bounds you might as well ignore the effect. At 2d6-10 or worse, while the assumed 2d6 value seems high the expected damage outcomes are low enough that you simply don't care. At 2d6-4 and better, you are getting close enough to 7 that the error made is small enough that you just don't care. Now, if you really want to taunt me into a huge wall of text, ask me about what "lucky" really means, and we can discuss Variance. -und_ed I would like to read this wall of text I think I have general idea how this works but never seen WM&H specific numbers for this.
Comming back to: 2d6-9: 9.3 Last number is expected dmg? Example is fine but I don't understand this list for different 2d6-X.
Yeah, so after all that on 2d6-9, your expected damage per attack is 0.3 It's not really meaningful until you're attacking with 10 or more attacks, and even then your expected damage is still pretty low (10 attacks at dice -9 expect 3 damage), so not really a big deal. Where it becomes a big deal is at 2d6-7 and 2d6-6, where your expected damage values become significantly different to the assumed average of 7. There are similar issues of course with 3d6-X and other permutations, but it takes a bit more time to get the numbers for those. As for the variance discussion, I'll bite on that when I have some more free time. I do tend to rant a bit on this topic, so please excuse the spittle. -und_ed
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Post by droopingpuppy on Sept 28, 2017 12:57:20 GMT
Let me start this by saying math is not my strong suit, so please correct me if I say something wrong. I was wondering if we could start a discussion on common dice math misconceptions. I rely on dice math pretty heavily in game, and I hear a lot of things that don't seem to make sense. For instance, if you are damaging something at dice -5, logic would suggest that you should average two damage per model that hits given an average dice roll of 7 and each higher and lower roll having a reciprocal dice roll at equal probability. But this discounts that everything 5 or below is treated equally, so a true average doesn't seem to work. In my mind you would need to take the average of all dice that are 6 or greater, average those, and then multiply that number times the number of times that you would roll over a 5. So if the average damage at dice -5 of all dice over a 5 are equal to 3.15 (according to my math), and you will roll over a 5 on 26 of 36 rolls, then 3.15 * (26/36) = 2.28. I know this isn't a huge difference, but logically the effect is amplified at every step. So at dice -6 you should average 1.55 instead of 1. This kinda thing can add up. If you attack with 10 infantry, you have roughly a 25% chance of rolling boxcars ((35/36)^10). What are some other math mistakes you see people make frequently in a game? What am I assuming in a game that is probably off? In the case, the answer would be 2.277 as you suspect, because; Roll probability damage 2 1/36 03 2/36 0 4 3/36 0 5 4/36 0 6 5/36 1 7 6/36 2 8 5/36 3 9 4/36 4 10 3/36 5 11 2/36 6 12 1/36 7 It would be all the possible probability occured if you rolled out 2d6 damage roll with -5 to damage. So, the average damage is (1x5+2x6+3x5+4x4+5x3+6x2+7x1)/36=2.27777..... -------------------------------- The thing you should be remember that nothing proves an inevitable success if you have a chance of failure. Even in 2+ to hit you misses if all dices are rolled 1. High probability means that it would be succeded on the higher chance, but there is some chance of failure. 97.2% to hit also means 2.7% to miss as well. But if you play many games, usually you will succeeded much more than failure in the similar situations, and it is why we are consider the probability. Regardless, if you think about boost the attack roll or make an additional melee attack by warjacks and warbeasts, and if you can hit 7+ or lower then you better off making additional melee attack. in 8+ it is worth considering to scaling the low chances to hit and the damage roll if you actually make a hit. in 9+ you better boost the attack roll anyways. If you really want to hit an attack then it is always better to boost the attack roll, but if you just want to cause more damage with your beatstick heavy and just wonder boost the attack roll or make the additional attack, then choose to make an additional melee attack roll is better in the most times.
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crimsyn
Junior Strategist
Posts: 389
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Post by crimsyn on Sept 28, 2017 14:22:35 GMT
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crimsyn
Junior Strategist
Posts: 389
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Post by crimsyn on Sept 29, 2017 5:02:45 GMT
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twity
Junior Strategist
Posts: 179
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Post by twity on Sept 29, 2017 12:11:10 GMT
Thanks, excel spreadsheets are always awesome. I miss the days when I could use them to analyze my data.
I liked your write-up a lot too, compounding probabilities are hard to calculate in your head and I think this did a nice job of showing that nice curve of probabilities. The only thing I would add is that it might be useful to show these probabilities with less information (as much as it pains me to say) if people want to use it as a quick reference sheet. It is pretty easy to multiply 2.09 in your head, but less so 2.08796. I realize people can round on their own and move on, but if you can eliminate one step in a game I think it is helpful.
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Post by Aegis on Sept 29, 2017 14:00:41 GMT
Yeah, is a common misconception, and is even more easily proved by this:
Using the common theory, infinite dice-7 2d6 damage rolls should do 0 damage (since 7- the median roll 7 = 0), but in practice is evident that the more you roll, the more dices will get results higher than 7 and the more damage will be done.
The averange damage of 100 rolls at 2d6-7 is far from zero, for the exact reason you explained, that all the results below 7 will still net "0" damage, not negative damage, while results higher than 7 will pile up without being "balanced" by the negatives of low rolls.
On the things of more dices: For sure the best way to use them is on things that can do multiple attacks. That said, sometimes piling up dices could be interesting exactly for the reason of the OP. More dices have chances to spike higher, and since high spikes count more than low spikes, in general more dices will net more damage against high armored targets than fixed numbers or low number of dices at parity of medium damage roll.
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zeffid
Junior Strategist
Posts: 163
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Post by zeffid on Sept 29, 2017 14:23:16 GMT
Dice probabilities are not very complex topic overall. The thing is that you need to do some calculations to get a proper result. That is why people use approximations in gaming. 2d6=7 is a good aproximation as long as you roll dice off one-two-three. It gets complicated when it is dice off 6 and one good roll will just ruin all the estimations. So these are the most common approximations that are used: 1) 2d6 = 7; 3d6=10.5; 4d6=14 - which helps to estimate to hit probability and occasionally damage rolls. 2) hand of fate (and like) give you a +2 - which is helpfull but rolling 2d6 for 11 under HoF is almost as hopeless as without
Going a bit more advanced you can estimate damage rolls (e.g. 3d6 - 10) as that 50% of rolls will produce damage and that can be averaged to 3 per roll. So 9 damage out of 6 hits.
All other calculations seem to be too hard for in-game usage.
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