Power Attack, Two Handed Weapons and Crit

[Staffan]

As an aside, you need to do pretty massive damage on a crit to justify using a weapon like the Falchion (though even then it is hard to justify)....though I do use the houserule of 1.5 for 2 handed and .5 for offhand when power attacking

Hey! That's my house rule!

 

[TerraDave]

the houserule of 1.5 for 2 handed and .5 for offhand when power attacking

Hey! That's my house rule!

It should be everybody's! oh wait, it shouldn't be a house rule at all...

 

[frisbeet]

but how much does this "lowest threat #" matter--it seems pretty restrictive.

But your broader point is well taken, and confirmed by my calculations above.

The point about oppenent type and world is also highly relevant (so be sure to carry that silvered undead bane great sword as well as the falchion)

The restriction to the said formula matters from the following derivation, which I will politely color orange for those less inclined to drool through algebra:

Suppose you’re fighting badguy X. It turns out you need, say, a 15 or better to hit X. You threaten a crit on an 18 since it turns out you’re wielding a Falchion. X is vulnerable to crits (aren’t we all).

The probability of hitting for crit damage =

3/20*6/20

An 18, 19, or 20 threatens out of 20 possible d20 outcomes, so P(threat) = 3/20. There are 6 ways of confirming the crit among 20 possible d20 outcomes, so P(confirm) = 6/20. One law of probability requires that when two things are needed for another thing to happen, the P(another thing happening) = P(thing1 happening)*P(thing2 happening). Ergo, 3/20 * 6/20.

The probability of hitting for normal damage =

3/20 + 3/20*14/20

That’s 3/20, the chance of rolling a 15, 16, or 17, plus 3/20*14/20, the chance of rolling an 18, 19, or 20, then missing the crit confirmation. That’s another law of probability: P(thing1 or thing2) = P(thing1) + P(thing2).

Now, generalize. Let N = # of d20 outcomes which hit-but-don’t-threat. This number can be 0, but not less than 0. Let C = threat range. For Falchion, C = 3.

Then P(hitting for crit) =

C/20*(C + N)/20

And P(hitting for normal) =

N/20 + C/20*(20 – N – C)/20

Ok, so what is an “automatic” hit? Isn’t P(hitting) = 1 for an automatic hit? Yes, but it doesn’t help us when we think about critical hits. Sometimes, those automatic hits land for a crit—but not always, of course. The proper way of thinking about it is: When I’ve tabulated all the hits I’ve made with a Falchion in my career, what % of the time were hits crits? The analytic answer to that is simply

P(hitting for crit) / {P(hitting for normal) + P(hitting for crit)}

Or

C/20*(C+N)/20 / {N/20 + C/20*(20 – N – C)/20 + C/20*(C+N)/20}

Algebra:

= (C^2 + C*N)/400 / (20*N + 20*C – C*N – C^2 + C^2 + C*N )/400

= (C^2 + CN) / (20N + 20C)

= C(C+N) / 20 (C+N)

= C/20

Holy schnikees, that seems obvious! Among automatic hits, the % of the time it’s a crit is the threat range / 20.

Deriving the rest of the formula I first gave:

Average damage/hit = P(hit not crit) * (D + db) + P(crit) * (wd + db) * M, where M is the weapon crit multiplier, wd the average weapon damage, and db the damage bonus.

Since we’re talking automatic hits, P(hit not crit) is = 1 – P(crit), not the P(hitting for normal) formula given above. This = 1 – C/20. Then

Average damage/hit = (1 – C/20)*(wd + db) + C/20*M*(wd+ db).

= (1 – C/20 + C/20*M)*(wd + db)

= (1 + C/20*(M-1))*(wd + db)

Which is what I offered in my second post.

Great, answer the question frisbeet. What condition did I require in this derivation? Answer: N, the # of d20 outcomes which hit-but-don’t-threat, must be >=0. This formula is exactly wrong when badguy X’s AC is high enough such that only a 20 or 19 or (generalizing) 20-C+2 or better hits. A very subtle point. Gotcha! for reading this.

Which is why, I think, it’s clearer to just do the calculations against several AC taking into account a whole bunch of things yada yada yada click sig.

 

[TerraDave]

The restriction to the said formula matters from the following derivation, which I will politely color orange for those less inclined to drool through algebra:

Suppose you’re fighting badguy X. It turns out you need, say, a 15 or better to hit X. You threaten a crit on an 18 since it turns out you’re wielding a Falchion. X is vulnerable to crits (aren’t we all).

SNIP.

Great, answer the question frisbeet. What condition did I require in this derivation? Answer: N, the # of d20 outcomes which hit-but-don’t-threat, must be >=0. This formula is exactly wrong when badguy X’s AC is high enough such that only a 20 or 19 or (generalizing) 20-C+2 or better hits. A very subtle point. Gotcha! for reading this.

Which is why, I think, it’s clearer to just do the calculations against several AC taking into account a whole bunch of things yada yada yada click sig.

Err, thanks. But those spreadsheets are way cool--and yes one of them allows a detailed comparison of your charecter with falchion vs. great sword.

 

[Nail]

The spreadsheets are quite cool; I've loved 'em ever since I first grabbed them....In fact I think that was almost 2 years ago now! Wow! That's like 100 years in internet time! Frisbeet, how long have you had these sheets out there?

 

[frisbeet]

Hey thanks.

Lessee, when did 3.0 come out? Few months after that So I guess spring of 2003.

I've always wanted to be more ambitious with the sheets, such that no matter what character you've got you can exactly model him/her. I didn't want to make a more generic ENTER AB. ENTER WEAPON DAMAGE. spreadsheet because it I wanted to reduce the amount of lookup stuff a user needed to do--one could just try whatever series of characters was interesting and see how badass they were. Unfortunately it quickly became apparent that all the supplemental stuff was going to make this hard to do. Actually the friggin monk with flurry of blows presented a real problem, and I'm not happy with this implementation (it's correct, but you can't stack TWF with flurry, for example). So anyways it's an amalgam of the user lookup/specific selection style.

The real gem of the sheet is "rolls", though it's seldom necessary. This only needs to come into play when the opponent has DR. DR screws up average damage because you can't do negative damage. This formum actually (re)derived a formula a couple years ago which calculated the # of ways, & hence the probability, of getting an outcome rolling N identical s-sided dice. Thus enabling the rolls worksheet. Of course, the damn thing is half the reason why the spreadsheets are so large.

Last regrets: haven't had the time to keep the Comparison spreadsheet current--though it's numbers are correct--as I've endeavored to update the Power Attack Calculator. I see the PA calculator as the real utility of the two spreadsheets, as few other resources on the net (none that I know of) actually give you exact answers on the subject. Anywho, thanks for the indulgence.