Hello everyone!
I've made a simple spreadsheet to calculate the odds of attacks hitting and critting. It takes into account the rules for +10/-10 crits/fumbles, and the fact that natural 20's and 1's modify the degree of success.
The spreadsheet can be found here. You can copy it to your own Google account and change the numbers on the second and third sheet to see what effects that has on the graphs (it will shift them left and right).
The spreadsheet includes a couple graphs. The first shows, for a given fixed attack bonus, your probability of hitting or critting against a range of AC's. An image of that is here for those who can't view the spreadsheet. To get the total chance to hit you would add these two results together at a given AC.
There is another graph included in the spreadsheet which is the inverse, with a fixed AC and a range of attack bonuses. But really it's the same graph mirrored, so the first graph conveys the same info. I included it in case that makes more sense to someone.
I named these graphs and sheets keeping in mind that the same math applies to skill checks, saves, and any other check against a DC in Pathfinder 2. I didn't include graphs with miss and fumble chances, but those could be added on as well.
In making these tools I hope to help others understand how bonuses to attack or AC affect the probabilities, and potentially fuel discussions about relative power levels and how much benefit attack or AC bonuses provide in context. I hope someone finds this useful or at least interesting!
Here's another graph, showing the fixed AC chart with an added in "total value" line. Basically, this line shows the number you multiply your average damage by to reach your expected damage for one attack against the target AC.
From this new line, we can see that there is a clear region where bonuses to attack translate to steeper gains in damage. This region corresponds to having an attack bonus 10 points under the target AC, up to 2 points under the AC. In this region the gains from an attack bonus are double the regions before and after it.
What does this mean practically? The most benefit for a bonus on your check is gained when your base roll needs between a 2 and 10 to meet the DC. So if you have two allies who do roughly the same damage, the first hits on a 12 and the second hits on a 10, it's better to give an attack buff to the one who hits on a 10, since the increase to their expected damage is double the increase the first ally would get. However, if you're fighting weak enemies where the first guy already hits on a 2 or higher (and further attack buffs will only change hits into crits), buffing him to increase his crit range will have less benefit than buffing the less accurate guy, who will benefit from an increased crit range and a decreased miss chance.
Also, if the two allies from the above example are fighting a boss enemy where their required rolls to hit are 16 and 14, you know that buffing either of them will have the same effect, since they both lie in the same region of the line. So we don't need to worry about our more accurate allies always being the go-to target for buffs, since the optimal buff target varies based on circumstance.
That is true for practical situations. And in most cases where the fighter is either too high or low to be in the "sweet spot", the Barbarian who is only two attack behind them will probably be similarly too high or low, so this wouldn't be a determining factor in who to buff. Only if a target AC is right on the edge would this matter, and as you say the PCs won't know the exact AC.
In the end the best default action is usually to buff the attack of the character who does the most damage on hit, since the proportional increase will usually result in the most damage gained. But we didn't really need this graph to know that.
It's less true in this edition that you won't know the AC. There are more ways to narrow it down than in PF1 where you hit on a 15, miss on a 14, and now know what it is.
Now you can do the above, plus crit without a 20 (know you're 10 over), crit fail without a 1 (know you're 10 under), see the opponent shield block (or parry depending on how your GM describes things) for no damage to know you're within 2. You can now narrow it down more quickly.
Thank you very much for putting together this chart!
Now that you've done it, could you add a weapon damage evaluator to it that includes number and size of dice, deadly/fatal, mods, and the works so that we can do a damage assessment vs. AC? I know it's a small request, but you're so amazing at this!
Seriously though, this was a great share. I did some crude similar comparisons to try to analyze light pick vs. 1d6/1d8 type weapons and found fatal d8 picks higher in damage. However, I can't remember the statistics rules very well, so it ended up being too manual to use for other comparisons. This is handy.
Now that you've done it, could you add a weapon damage evaluator to it that includes number and size of dice, deadly/fatal, mods, and the works so that we can do a damage assessment vs. AC? I know it's a small request, but you're so amazing at this!
Yes, this was the intention for what to do with it after this but figured this would be useful to others trying to do similar calculations. This chart's main purpose for me was to try and find that "total value" equation (which is of course a set of equations), so now that can be used in further spreadsheets instead of manually calculating hit and crit chances.
Figuring out how that next spreadsheet will look is going to be tricky with so many options to analyze, and this analysis is likely going to become multidimensional so figuring out some way of doing that in spreadsheets and graphing it will be tricky as well. Likely I'll try to start with analysis of a smaller set of options, like looking at which weapon works best against which ACs for a 2h fighter (of course quantifying non-damaging weapon traits is another issue).
It gets complex. What are assumptions? Probably 3 attacks per turn, but that's complicated by power attack, so maybe a 3 attack matrix to begin. That's complicated by agile and forceful and doesn't include sweep, so where to begin?
For my light pick analysis, I assumed 3 attacks and then did a hit chart much like yours assuming averages for damage. Add ratio of normal hit damage to crit damage for comparison. The light pick crit damage was so much higher that it turned out always superior to the long sword further helped by agile (assuming ya know maths).
A chart like that could be a start. Then you could add on check boxes for things like agile and tweak the end calc formula to factor them or not. Some of it could just be left to the viewer's imagination to infer advantages I'm sure. You could spend all day on this stuff.
Here's the reults for the "total value" equation (anyone have a better name for the combination of hit and crit?), so that it can be used by me and others for further analysis:
For an attack bonus B and an AC A, the hit chance multiplier for average damage is given by the following set of equations (sorry for improper notation, it's a forum and I haven't taken a math class in a little while :P):
if A - B <= -9 ===> 1.95,
if -9 < A - B <= 2 ===> 1.95-0.05(9+A-B),
if 2 < A - B <= 10 ===> 1.4-0.1(-2+A-B),
if 10 < A - B <= 21 ===> 0.6-0.05(-10+A-B),
if 21 < A - B <= 30 ===> 0.05,
if 30 < A - B ===> 0
These can be summarized in the Excel equation that follows, given AC is cell A2 and hit bonus is cell B2:
=IF(A2-B2<=-9, 1.95, IF(A2-B2<=2, 1.95-0.05*(9+A2-B2), IF(A2-B2<=10, 1.4-0.1*(-2+A2-B2), IF(A2-B2<=21, 0.6-0.05*(-10+A2-B2), IF(A2-B2<=30, 0.05, 0)))))
Of course this combined hit and crit, and assumes crits don't add extra value over x2 damage. So they would need to be broken apart instead with your analysis of deadly weapons.
One thing I think is an interesting result here, though not sure if it has any practical use, is that the point where you are expected to do 1x weapon damage per attack is when your attack bonus is 6 lower than the target AC. So hit on a 6 and crit on a 16.
I'll probably do similar equations for attack and crit independently, so that bonus crit effects can be taken into consideration.
After thinking about these results and the discussion some more, and which direction I can build next, I think my next avenue will be to input a fixed attack bonus and average damage per hit for two characters, along with a fixed attack buff value, and have the end result tell you which character would be better to buff against a range of ACs (including how much damage is gained by either character). This will let us explore where the "tipping points" are for these moving parts, and better inform the spell caster's decisions.
For example, you should be able to find the point where casting your own damage spell would theoretically be more valuable than a buff spell (given a bunch of constraints), which is important to know. This would let us know whether the claims (exaggerating here) that casters are useless except as support for the martials are true, or what situations that is more or less true.
Of course, and the more attacks a martial can make with a buff the more average damage that spell has contributed, so you need to know how many attacks can be expected as well. If any conclusions can be drawn they will have to be heavily constrained.
More graphs! These are added to the spreadsheet linked in the original post.
I'm starting with a very simple comparison that's been done a number of times already, the level 1 two-handed fighter against the level 1 two-handed barbarian. The fighter has a +9 to hit (+4 expert, +1 level, +4 STR) for 10.5 damage (1d12+4). The barbarian has a +7 to hit (+2 trained, +1 level, +4 STR) for 12.5 damage (1d12+6). Both are using a d12 weapon with no traits that affect this comparison (so greatsword, maul, or greataxe) and we are currently only comparing attacks with full bonus, no MAP.
As we can see from the first graph, fighter is typically ahead in this comparison. Below AC 14 the barbarian does more damage, but most enemies even at level 1 have higher AC than that. They get very close at AC 19-20 (due to the "sweet spot" where fighter loses more per AC mentioned in my previous post) but fighter stays on top until AC 30 (which is higher than you will be encountering at that level).
The second graph is using a fixed buff of +1 to hit on both characters, which could be a bless spell or an inspire courage, and shows how much difference in damage that fixed buff gives. It shows that buffing the barbarian is the best choice, except at AC 18-19 and AC 29-30. This is what we expected from previous analysis, but this graph can yield some more interesting results.
This third graph shows what happens when we change the +1 bonus to a +3. Now, we can see that the sharp angles are flattening, and the fighter line has a wider range of AC values for which it's above the barbarian line. In fact, it now covers four values for both ranges, at 18-19-20-21 and 29-30-31-32. This shows that as the buffs get bigger our assumption of always buffing barbarian may not hold. It suggests further investigation may be needed, especially for higher levels.
Finally, I would like to point at the added damage on the second graph, which is the Y axis. We can see that our first level barbarian's basic Strikes don't actually gain that much benefit from our buff, at between 1.25 and 0.625 added average damage per attack for enemies we might fight at this level (probably 1.25 for their first attack and 0.625 for their second, just move 5 to the right on the graph for MAP). So we can put the expected damage increase for a +1 attack on our barbarian for the combat somewhere around 7.5 damage (four rounds of attacking two times per round). Most characters will be using attacks which do less damage on hit then that, and thus will have smaller added damage numbers.
This still isn't nearly enough to get any kind of conclusion out of it. Any martial character is going to have weapon traits, class features, and special attack actions which will change these base assumptions and graphs. But it's at least a starting point in evaluating buffs. I appreciate any feedback!