D&D 5E House rule: Extra Attacks for martials at 5,11,17

[Kinematics]

Your math is a nice simple approximation, but it is simply not correct for calculating the actual expected value. That's why I got a different result compared to doing the simple calculation you did above--because the number of crits you can potentially get depends on how many hits you already got (since you cannot crit if you missed!)

Not correct. There is a correlation between hits and crit (though I wouldn't say a dependency), but the individual attacks are completely unrelated to each other. There is no dependency between attacks.

If you make one attack each turn, then after four turns you have exactly the same damage distribution as if you made four separate attacks on one turn. The probability of any individual attack does not influence the probability of any other attack, nor does the chance of hitting have any bearing on the crit totals (though it will influence what percentage of your total damage comes from crits). So all you need to do is examine the damage results of a single attack and multiply that by 4.

You can, of course, examine how the number of crits are distributed if you want.

0 crits: 95% * 95% * 95% * 95% * 1 combination = 81.450625%
1 crit: 5% * 95% * 95% * 95% * 4 combinations = 17.1475%
2 crits: 5% * 5% * 95% * 95% * 6 combinations = 1.35375%
3 crits: 5% * 5% * 5% * 95% * 4 combinations = 0.0475%
4 crits: 5% * 5% * 5% * 5% * 1 combination = 0.000625%

And then use that to calculate what the average amount of crit damage you'll see over 4 attacks would be:

Average crit damage = 5.5

81.450625% * 5.5 * 0 + 17.1475% * 5.5 * 1 + 1.35375% * 5.5 * 2 + 0.0475% * 5.5 * 3 + 0.000625% * 5.5 * 4
= 0 + 0.9431125 + 0.1489125 + 0.0078375 + 0.0001375
= 1.1

Alternatively: 4 attacks * 5% crit chance * 5.5 damage per crit = 1.1

 

[EzekielRaiden]

To be clear, none of this is with the goal of balancing anything.

Right. You asked if it could push those jobs past Paladin. My gut feeling response, without proper testing, is "no, unless you ensure that the 5MWD pattern is prevented on the regular." That is a balance-adjacent argument, since 5MWD is a balance concern, but it's also just...a comparison of things. A Barbarian that gets starved of daily fights (rather, combat rounds per long rest) won't be able to push past the Paladin for overall power. You can't compare them without factoring in that Paladins are more daily-focused than other "warrior" types. (Barbs are too, since Rage is uses per day, but you get to 4/day very quickly.)

 

[EzekielRaiden]

Not correct. There is a correlation between hits and crit (though I wouldn't say a dependency), but the individual attacks are completely unrelated to each other. There is no dependency between attacks.

If you make one attack each turn, then after four turns you have exactly the same damage distribution as if you made four separate attacks on one turn. The probability of any individual attack does not influence the probability of any other attack, nor does the chance of hitting have any bearing on the crit totals (though it will influence what percentage of your total damage comes from crits). So all you need to do is examine the damage results of a single attack and multiply that by 4.

You can, of course, examine how the number of crits are distributed if you want.

0 crits: 95% * 95% * 95% * 95% * 1 combination = 81.450625%
1 crit: 5% * 95% * 95% * 95% * 4 combinations = 17.1475%
2 crits: 5% * 5% * 95% * 95% * 6 combinations = 1.35375%
3 crits: 5% * 5% * 5% * 95% * 4 combinations = 0.0475%
4 crits: 5% * 5% * 5% * 5% * 1 combination = 0.000625%

And then use that to calculate what the average amount of crit damage you'll see over 4 attacks would be:

Average crit damage = 5.5

81.450625% * 5.5 * 0 + 17.1475% * 5.5 * 1 + 1.35375% * 5.5 * 2 + 0.0475% * 5.5 * 3 + 0.000625% * 5.5 * 4
= 0 + 0.9431125 + 0.1489125 + 0.0078375 + 0.0001375
= 1.1

Alternatively: 4 attacks * 5% crit chance * 5.5 damage per crit = 1.1

Then what was wrong with my numbers, which came from the things you're referencing here? I broke it down differently, but the numbers are there (e.g. the 0.171 probability thing), and yet produced 2.18 as the final result?

 

[TwoSix]

Not correct. There is a correlation between hits and crit (though I wouldn't say a dependency), but the individual attacks are completely unrelated to each other. There is no dependency between attacks.

Exactly. Don't break out binomial distributions for independent events.

 

[TwoSix]

Right. You asked if it could push those jobs past Paladin. My gut feeling response, without proper testing, is "no, unless you ensure that the 5MWD pattern is prevented on the regular." That is a balance-adjacent argument, since 5MWD is a balance concern, but it's also just...a comparison of things. A Barbarian that gets starved of daily fights (rather, combat rounds per long rest) won't be able to push past the Paladin for overall power. You can't compare them without factoring in that Paladins are more daily-focused than other "warrior" types. (Barbs are too, since Rage is uses per day, but you get to 4/day very quickly.)

Sure. Again to be clear, I'm actively interested on what the balance impact IS, but the goal of the change is not around achieving balance.

So basically a 1-2 encounter day would favor paladins still once they go nova-happy, 3-4 might favor barbs (since they can still rage but paladins are running out of smite juice), and any more will probably favor barbs on offense but paladins are still defensive/support gods with much higher AC.

And of course, pala 2 and/or pala 6 will still be ideal for MCing, nothing here changes that.

 

[TwoSix]

Then what was wrong with my numbers, which came from the things you're referencing here? I broke it down differently, but the numbers are there (e.g. the 0.171 probability thing), and yet produced 2.18 as the final result?

Just at a quick glance, I think the probabilities of critting once hit rate is factored in are wrong.

The obvious example is the 4-crit case, where you have "0.1785×0.05^4×22". We know the odds of getting 4 crits is exactly 1/(20^4), so the probability calculation for crits must be wrong. Since the 0.1785 (the 4 hit probability) assumes all the use cases where a d20 has a value of 1-7 are eliminated, the probability in your formula should be 0.1785 x (1/(13^4)) x 22, which works out to be right about 1/20^4 (or 1 in 160000).

That bumps to total value up. I'm thinking some of the probability of the use cases in the multi-miss scenario probably need to be bumped down, since they probably aren't accounting for the missing dice.

 

[Kinematics]

Then what was wrong with my numbers, which came from the things you're referencing here? I broke it down differently, but the numbers are there (e.g. the 0.171 probability thing), and yet produced 2.18 as the final result?

This is the (simplified) formula you used:

0.1115×0.05×5.5 + 0.3105×(0.05^2×11+0.975×5.5) + 0.3845×(0.05^3×16.5+0.0071×11+0.1354×5.5) + 0.1785×0.05^4×22+0.00048125×16.5+0.0135×11+0.1715×5.5) = 2.21

Base accuracy: 65%
Miss rate: 35%

Chance of landing 0 hits out of 4 attacks = 0.01500625 (rounded to 0.0150, or 1.50%) (not used above)
Chance of landing 1 hit out of 4 attacks = 0.111475 (rounded to 0.1115, or 11.15%)
Chance of landing 2 hits out of 4 attacks = 0.3105375 (rounded to 0.3105, or 31.05%)
Chance of landing 3 hits out of 4 attacks = 0.384475 (rounded to 0.3845, or 38.45%)
Chance of landing 4 hits out of 4 attacks = 0.17850625 (rounded to 0.1785, or 17.85%)

Then look at some of the components. For example:

0.1115×0.05×5.5

This is the 11.15% chance of landing 1 hit out of 4, the 5% chance of landing a crit, and the 5.5 damage the crit did. The problem here is that the 5% * 11.15% calculation is not correct. You've already landed your one attack, but multiplying by 5% indicates that only one out of twenty of those hits was a crit. This is incorrect.

Put in die roll terms, a 65% hit rate means that you rolled an 8 or better. The 35% miss rate is rolling a 1 through 7. That means there were 13 possible values for you to have rolled, of which one of them is a crit.

Thus, given that you know you already hit, the chance of it being a crit is 1 in 13, not 1 in 20.

Of course this would mean that you're undervaluing how much crit damage was done, rather than overvaluing as seen in the actual result, so let's continue to the next.

0.3105×(0.05^2×11+0.975×5.5)

This one is landing two hits, and then tries to combine the value of both hits being crits, and the chance of one hit being a crit. There's still the 1 in 20 vs 1 in 13 error, but in addition to that you've assumed that if you didn't land two crits (0.05^2), that you are almost guaranteed to have landed one (0.975).

I'm not sure where you got the 0.975 from. It looks like perhaps a typo versus the 0.9975 that would be all combinations that were not two crits. Even if we keep the assumption of 5% for the crit, this value should be 0.095. This creates a 1.5 point overvaluation before correcting to 1/13.

0.3845×(0.05^3×16.5+0.0071×11+0.1354×5.5)

The values here are correct for a 1 in 20, but of course need to be corrected to 1 in 13.

0.1785×(0.05^4×22+0.00048125×16.5+0.0135×11+0.1715×5.5)

0.00048125 appears to be incorrect. It should be 0.000475.

So overall you need to subtract 1.5 from the 2.2, leaving you 0.7, but then everything needs to scale up because you should be using 7.69% (1 in 13) instead of 5% (1 in 20).

You'd end up with:

0.1115 * (0.0769 * 5.5)
+ 0.3105 * (0.0769^2 * 11 + 0.1420 * 5.5)
+ 0.3845 * (0.0769^3 * 16.5 + 0.0164×11 + 0.1966×5.5)
+ 0.1785 * (0.0769^4 * 22 + 0.0004198×16.5 + 0.0302×11 + 0.242×5.5)
= 0.1115 * 0.423
+ 0.3105 * 0.846
+ 0.3845 * (0.0075 + 0.1801 + 1.0812)
+ 0.1785 * (0.000769 + 0.00693 + 0.3326 + 1.331)
= 1.096

Which, allowing for rounding (I only used the approximate of 1/13), matches the expected 1.1.