Bear in mind that I'm a liberal arts major, so math isn't my strong suit. But I think it would just be:
[normal damage forumla] + ([chance of one miss] × [normal damage formula])
I think that should be accounted for because the number of hits in the normal formula is just a percentage.
Say you have 1 shot at BS4. Every shot you should get 0.5 hits and 0.5 misses. Throw in balanced, and every shot you get 0.5 hits and 0.5 rerolls for a total of 0.75 hits. Again, I'm not sure if it's right, but that's how I handled it when I made a similar spreadsheet at the start of the new edition.
Edit: I think I see what you're saying, and my response doesn't really address it. I'll have to think about how to manage the chance of one miss formula.
You want the probability of *at least* one miss. The probability of getting a nonzero number of misses is 1.0 minus the probability of getting exactly zero misses (all hits). Hitting on a 2+ and rolling 4 dice, the probability of all hits is (5/6)^4 which is roughly 48%, so a roughly 52% chance of at least one miss.
[expected damage of N dice, with Balanced] = ([expected damage of one die without rerolls] * [N]) + ((1.0 - [probability of zero misses]) * [expected damage of one die without rerolls])
I ***approximate*** dice calculations in my head. Hopefully the logical process is useful.
The crux of the problem is you need to get to get more hits than an opponent's defense. So, what you actually care about is number of successes. And, as a basic feature, both players can critical strike. However, because the defender can spend two successes to negate a crit, we can approximate that they're interchangeable.
Therefore, rolling a 6 is basically like scoring two successes. Like exploding dice in big 40k, statistically that's the same as +1 to hit (or save).
So a 3+ Boltgun rolls 4 dice and scores 0.83 average successes per dice for 3.32. A Guardsmen in cover retains 1 success, then rolls two more at 0.5 average, for an average of 2 successful defense dice. Which means you get an average of 1.32 attacks passed their defense. Multiply that by your damage and you get 3.96 average damage. If you check with a dice simulator, the real average is 4.05, so we got within 3% by doing simple math in our head.
Here's a fun example. My Craftworld Dire Avengers have 4A BS3+ 3/4 with Balanced and Rending. Well, 80% of the time, you miss at least once on BS3+. So, Balance is like having 0.8 extra attack rolls.
Similarly, Rending just means your first crit is worth 3 successes (2 for itself, and it adds a success to another hit by upgrading it). That means having Rending is like an extra +1 to hit. In edge cases where you roll a lot of crits then it is worth less but we're doing simple math.
So, the Dire Avenger essentially rolls 4.8 dice, and each dice on average scores 1 success. That gives 4.8 average hits, against a Guardsman's 2 defense in cover, for 2.8 successes. We know in practice you're not actually getting 3 more hits. What this usually means is that you scored a crit and slightly less than a hit, and so dealt an average of 4+(0.8\*3)=6.4 damage. The dice simulator says the real average is 6.48, so we're 98% accurate.
I can be specific and do the same approximation for your Vet Guard Sniper.
A good way to think about Mortal Wounds is you score guaranteed damage based on your chance to crit. For simplicity, the math works out to 2/3s average damage per MW#, so MW3 is 2 extra damage.
In this case, we don't care about crits counting as 2 successes. We know hitting on 2s likely means we'll score a couple more hits than their defenses. Crits being harder to block matters a lot less anytime there's a large difference of attack dice to defense, like flamers.
So, if we say the real number of successful defense dice for the Guardsmen is 1+0.33+0.33=1.67, and the real number of average hits for the sniper is 4\*0.83=3.33. That means 1.67 average hits get through, so multiply by 3 damage for 5 average damage. Then add the 2 extra mortal wounds, for 7 average damage. If you check the dice simulator, it says 7.19, so we're 97% accurate.
If you want to add Balanced, consider how often you're going to use the re-roll. You have about a 50% chance to miss at least once with 4 rolls at BS2+. Consequently, you get 0.5\*0.67=0.33 more Mortal Wounds.
Similarly, you get 4.5 rolls at BS2+ is 3.7 hits. Against 1.67 average defense rolls, you get about 2.07 hits for 3 damage each. So, 6.21 average damage + 2.33 average mortal wounds is 8.54. The dice simulator says 8.64, so we're 98% accurate.
You've forgotten to multiply the second term (the chance of getting at least one miss) times the expected damage of the die you get to reroll. You also are using the chance of one die missing instead of one die hitting. It worked in the simple examples you made because the expected damage of the die was 1 so multiplying it doesn't do anything, and the hit chance and miss chance were both 50%, but the Vet Guard Sniper's expected damage per die is more than 1 and hit chance is not 50% so your formula disagrees. Corrected version of the 4-dice formula:
( (4/6) * 3 + (1/6) * (3+3) ) * 4 + ( ( 1 - ( (5/6)^4 ) ) * ( (4/6) * 3 + (1/6) * (3+3) ) ) = 13.55
Kind of. 1 - ( (1/6)^4 ) is the probability of getting at least one non-crit, but just swapping that into the above formula does *not* give the expected damage of a Balanced attack where you reroll any non-crit, because you would have to account for the fact that rerolling a hit decreases your damage. With rerolls you generally need to subtract the damage of the result on the die you chose to reroll and add the expected damage of it after rerolling, but generally you only reroll misses so the subtracted term is just zero and thus can be ignored. This gets complicated because I assume you would reroll a miss if available and only reroll a normal hit if there were no misses, so you'd need to account separately for the case where you have at least one miss and the case where you have no misses but still at least one normal hit. I think the formula would be:
normalDamage = 3
pNormal = 4/6
critDamage = 6
pCrit = 1/6
expectedDieDamage = ( pNormal * normalDamage + pCrit * critDamage ) = 3
numDice = 4
pNonMiss = 5/6
pAnyMisses = ( 1 - ( pNonMiss^numDice ) )
pAllCrits = pCrit^numDice
pNoMissesAnyNormals = 1 - pAnyMisses - pAllCrits
rerollMissDamageBoost = expectedDieDamage
rerollNormalDamageBoost = expectedDieDamage - normalDamage
rerollNonCritDamage = expectedDieDamage * numDice + pAnyMisses * rerollMissDamageBoost + pNoMissesAnyNormals * rerollNormalDamageBoost = 3 * 4 + ( 1 - ( (5/6)^4 ) ) * 3 + ( 1 - ( 1 - (5/6)^4 ) - ( (1/6)^4 ) ) * (3 - 3) = 13.55
This gets the same value because the last term is zero: the expected damage of a rolled die is 3, and a normal hit does 3 damage, so rerolling it has no effect on the total amount of damage expected to be presented. Of course, because crits are harder to block you probably do actually see a slight boost here, but the math for that is complicated (and I haven't thought about how to go about it).
Rerolling up to a fixed number of dice also seems a bit complicated, though being careful you could probably do it. For 3 rerolls you could calculate the chance of getting exactly one miss times 1, plus the chance of exactly two misses times 2, plus the chance of *at least* three misses times 3, and this gives you the expected number of rerolls you'll get to do, which you then multiply by the expected damage per die. Rerolling N non-crits would be even more complicated.
You'd need a weapon profile where the expected damage is greater than the normal hit damage. For example, the Legionary Chosen's Daemon Blade hits on 2+, damage 4/7, Lethal 5+, so the expected damage is:
( (3/6) * 4 + (2/6) * 7 ) = 4.333 which is greater than the normal damage of 4, meaning rerolling a normal hit is worth it on average.
I think the best way to break this down is to sanity check as you go.
You start with 100% then you figure out the odds of different possibilities (which should add to 100%) and assign the expected value (in this case damage) of each possibility.
For example, there is a certain possibility that on 3 dice, you roll:
All misses
1 hit
2 hits
3 hits
1 hit 1 crit
1 hit 2 crits
2 hits 1 crit
1 crit
2 crits
3 crits
This covers the space of all possibilities for 3 dice, and the odds of them all should add to 1. The same is true for more dice, and the calculation is very similar. The same is true for balanced as well, it gets added possible outcomes for the possibilities where you don't get a total of 3 hits.
You can also seperately calculate the damage value for each possibility, and multiply that by the probability of each outcome to get an "expected damage" for an attack.
Note that for this to be accurate, you'll need to take into account save rolls, and the matrix of possibilities gets rather complicated then, but still solvable by a computer using programming or even something like excel
Your failed hit doesn't take anything into account. It is not a chance of anything. You will always get 0, because you multiply by 0, and the chance of a failed hit resulting in no damage will not be 0%.
I actually think you are calculating odds and not chance, but I can't really grasp your formula.
I think you are definitely on the right track with calculating probabilities for attacks in your calculator. However, I do agree with the sentiment that the chance of a failed hit resulting in no damage is not 0%. Also, I think the issue with representing "Balanced" in your calculator is a bit tricky and may require some creative thinking. Keep working on it though, and I'm sure you'll figure it out!
>I wanted to be able to compare multiple weapon profiles/modifiers next to one another and the site only does one at a time.
Try out https://denampavel.shinyapps.io/KTSim/
Also, you can fork https://github.com/jmegner/KT21Calculator and do your own changes. I would love to get a pull request on this.
Yes, doing precise calculations is hard, so that is why everybody else does Monte Carlo simulations. The hits minus saves (with crits as well) is especially hard to do analytically, so ktcalc actually has to do the cancelling stuff over each combo of attack and defense dice outcomes.
Wh40k math is much simpler because it is multiplicative in nature.
Bear in mind that I'm a liberal arts major, so math isn't my strong suit. But I think it would just be: [normal damage forumla] + ([chance of one miss] × [normal damage formula])
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I think that should be accounted for because the number of hits in the normal formula is just a percentage. Say you have 1 shot at BS4. Every shot you should get 0.5 hits and 0.5 misses. Throw in balanced, and every shot you get 0.5 hits and 0.5 rerolls for a total of 0.75 hits. Again, I'm not sure if it's right, but that's how I handled it when I made a similar spreadsheet at the start of the new edition. Edit: I think I see what you're saying, and my response doesn't really address it. I'll have to think about how to manage the chance of one miss formula.
You want the probability of *at least* one miss. The probability of getting a nonzero number of misses is 1.0 minus the probability of getting exactly zero misses (all hits). Hitting on a 2+ and rolling 4 dice, the probability of all hits is (5/6)^4 which is roughly 48%, so a roughly 52% chance of at least one miss. [expected damage of N dice, with Balanced] = ([expected damage of one die without rerolls] * [N]) + ((1.0 - [probability of zero misses]) * [expected damage of one die without rerolls])
I ***approximate*** dice calculations in my head. Hopefully the logical process is useful. The crux of the problem is you need to get to get more hits than an opponent's defense. So, what you actually care about is number of successes. And, as a basic feature, both players can critical strike. However, because the defender can spend two successes to negate a crit, we can approximate that they're interchangeable. Therefore, rolling a 6 is basically like scoring two successes. Like exploding dice in big 40k, statistically that's the same as +1 to hit (or save). So a 3+ Boltgun rolls 4 dice and scores 0.83 average successes per dice for 3.32. A Guardsmen in cover retains 1 success, then rolls two more at 0.5 average, for an average of 2 successful defense dice. Which means you get an average of 1.32 attacks passed their defense. Multiply that by your damage and you get 3.96 average damage. If you check with a dice simulator, the real average is 4.05, so we got within 3% by doing simple math in our head. Here's a fun example. My Craftworld Dire Avengers have 4A BS3+ 3/4 with Balanced and Rending. Well, 80% of the time, you miss at least once on BS3+. So, Balance is like having 0.8 extra attack rolls. Similarly, Rending just means your first crit is worth 3 successes (2 for itself, and it adds a success to another hit by upgrading it). That means having Rending is like an extra +1 to hit. In edge cases where you roll a lot of crits then it is worth less but we're doing simple math. So, the Dire Avenger essentially rolls 4.8 dice, and each dice on average scores 1 success. That gives 4.8 average hits, against a Guardsman's 2 defense in cover, for 2.8 successes. We know in practice you're not actually getting 3 more hits. What this usually means is that you scored a crit and slightly less than a hit, and so dealt an average of 4+(0.8\*3)=6.4 damage. The dice simulator says the real average is 6.48, so we're 98% accurate.
I can be specific and do the same approximation for your Vet Guard Sniper. A good way to think about Mortal Wounds is you score guaranteed damage based on your chance to crit. For simplicity, the math works out to 2/3s average damage per MW#, so MW3 is 2 extra damage. In this case, we don't care about crits counting as 2 successes. We know hitting on 2s likely means we'll score a couple more hits than their defenses. Crits being harder to block matters a lot less anytime there's a large difference of attack dice to defense, like flamers. So, if we say the real number of successful defense dice for the Guardsmen is 1+0.33+0.33=1.67, and the real number of average hits for the sniper is 4\*0.83=3.33. That means 1.67 average hits get through, so multiply by 3 damage for 5 average damage. Then add the 2 extra mortal wounds, for 7 average damage. If you check the dice simulator, it says 7.19, so we're 97% accurate.
If you want to add Balanced, consider how often you're going to use the re-roll. You have about a 50% chance to miss at least once with 4 rolls at BS2+. Consequently, you get 0.5\*0.67=0.33 more Mortal Wounds. Similarly, you get 4.5 rolls at BS2+ is 3.7 hits. Against 1.67 average defense rolls, you get about 2.07 hits for 3 damage each. So, 6.21 average damage + 2.33 average mortal wounds is 8.54. The dice simulator says 8.64, so we're 98% accurate.
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You've forgotten to multiply the second term (the chance of getting at least one miss) times the expected damage of the die you get to reroll. You also are using the chance of one die missing instead of one die hitting. It worked in the simple examples you made because the expected damage of the die was 1 so multiplying it doesn't do anything, and the hit chance and miss chance were both 50%, but the Vet Guard Sniper's expected damage per die is more than 1 and hit chance is not 50% so your formula disagrees. Corrected version of the 4-dice formula: ( (4/6) * 3 + (1/6) * (3+3) ) * 4 + ( ( 1 - ( (5/6)^4 ) ) * ( (4/6) * 3 + (1/6) * (3+3) ) ) = 13.55
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Kind of. 1 - ( (1/6)^4 ) is the probability of getting at least one non-crit, but just swapping that into the above formula does *not* give the expected damage of a Balanced attack where you reroll any non-crit, because you would have to account for the fact that rerolling a hit decreases your damage. With rerolls you generally need to subtract the damage of the result on the die you chose to reroll and add the expected damage of it after rerolling, but generally you only reroll misses so the subtracted term is just zero and thus can be ignored. This gets complicated because I assume you would reroll a miss if available and only reroll a normal hit if there were no misses, so you'd need to account separately for the case where you have at least one miss and the case where you have no misses but still at least one normal hit. I think the formula would be: normalDamage = 3 pNormal = 4/6 critDamage = 6 pCrit = 1/6 expectedDieDamage = ( pNormal * normalDamage + pCrit * critDamage ) = 3 numDice = 4 pNonMiss = 5/6 pAnyMisses = ( 1 - ( pNonMiss^numDice ) ) pAllCrits = pCrit^numDice pNoMissesAnyNormals = 1 - pAnyMisses - pAllCrits rerollMissDamageBoost = expectedDieDamage rerollNormalDamageBoost = expectedDieDamage - normalDamage rerollNonCritDamage = expectedDieDamage * numDice + pAnyMisses * rerollMissDamageBoost + pNoMissesAnyNormals * rerollNormalDamageBoost = 3 * 4 + ( 1 - ( (5/6)^4 ) ) * 3 + ( 1 - ( 1 - (5/6)^4 ) - ( (1/6)^4 ) ) * (3 - 3) = 13.55 This gets the same value because the last term is zero: the expected damage of a rolled die is 3, and a normal hit does 3 damage, so rerolling it has no effect on the total amount of damage expected to be presented. Of course, because crits are harder to block you probably do actually see a slight boost here, but the math for that is complicated (and I haven't thought about how to go about it). Rerolling up to a fixed number of dice also seems a bit complicated, though being careful you could probably do it. For 3 rerolls you could calculate the chance of getting exactly one miss times 1, plus the chance of exactly two misses times 2, plus the chance of *at least* three misses times 3, and this gives you the expected number of rerolls you'll get to do, which you then multiply by the expected damage per die. Rerolling N non-crits would be even more complicated.
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You'd need a weapon profile where the expected damage is greater than the normal hit damage. For example, the Legionary Chosen's Daemon Blade hits on 2+, damage 4/7, Lethal 5+, so the expected damage is: ( (3/6) * 4 + (2/6) * 7 ) = 4.333 which is greater than the normal damage of 4, meaning rerolling a normal hit is worth it on average.
Sorry, I was being a jerk. I only mention in 'in my head' because that's when I'd do the calculations manually: at the board, during the game.
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Hell, my message right now is still demeaning. Let me just tweak that. I don't know what I was thinking. I agree with Luminite's result.
I think the best way to break this down is to sanity check as you go. You start with 100% then you figure out the odds of different possibilities (which should add to 100%) and assign the expected value (in this case damage) of each possibility. For example, there is a certain possibility that on 3 dice, you roll: All misses 1 hit 2 hits 3 hits 1 hit 1 crit 1 hit 2 crits 2 hits 1 crit 1 crit 2 crits 3 crits This covers the space of all possibilities for 3 dice, and the odds of them all should add to 1. The same is true for more dice, and the calculation is very similar. The same is true for balanced as well, it gets added possible outcomes for the possibilities where you don't get a total of 3 hits. You can also seperately calculate the damage value for each possibility, and multiply that by the probability of each outcome to get an "expected damage" for an attack. Note that for this to be accurate, you'll need to take into account save rolls, and the matrix of possibilities gets rather complicated then, but still solvable by a computer using programming or even something like excel
🙄
Your failed hit doesn't take anything into account. It is not a chance of anything. You will always get 0, because you multiply by 0, and the chance of a failed hit resulting in no damage will not be 0%. I actually think you are calculating odds and not chance, but I can't really grasp your formula.
I think you are definitely on the right track with calculating probabilities for attacks in your calculator. However, I do agree with the sentiment that the chance of a failed hit resulting in no damage is not 0%. Also, I think the issue with representing "Balanced" in your calculator is a bit tricky and may require some creative thinking. Keep working on it though, and I'm sure you'll figure it out!
Create a simulator instead of a calculator. Seems easier. Math is hard.
>I wanted to be able to compare multiple weapon profiles/modifiers next to one another and the site only does one at a time. Try out https://denampavel.shinyapps.io/KTSim/ Also, you can fork https://github.com/jmegner/KT21Calculator and do your own changes. I would love to get a pull request on this.
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Yes, doing precise calculations is hard, so that is why everybody else does Monte Carlo simulations. The hits minus saves (with crits as well) is especially hard to do analytically, so ktcalc actually has to do the cancelling stuff over each combo of attack and defense dice outcomes. Wh40k math is much simpler because it is multiplicative in nature.