Critical Failure - twice as likely as Critical Success

[Ascalaphus]

I don't see a fairness issue, these rules apply equally to monsters and players and players fight monsters of both higher and lower level than them.

I also don't see a lack of clarity issue. The rules are pretty clear.

The issue is that some people don't like how viewed from one direction, it looks asymmetrical. The counterpoint to that is that viewed from the other direction, it is neatly symmetrical.

Not symmetrical:
0-9 Success (10 numbers)
-1 to -9 Failure (9 numbers)

But symmetrical, and IMO more relevant in practice, for any arbitrary DC like DC 23:

33+ Critical success
13- critical failure

In our decimal number system it's really easy to see if a number is 10 more or less than another number. With the current rules exploit this feature symmetrically in both directions.

This ease of use is far more important than abstract mathematical symmetry with no practical applications.

[Temperans]

I mean there is a practical application in that there is a higher chance of success.

Also the fact that its symmetrical with Critical but not with the overall roll does have some effect on things. But they are very insignificant overall.

[Malk_Content]

There is only a "higher chance of success" if the devs didn't account for this minor asymmetry when determining Target Numbers vs Expected Modifier groups. I doubt this is the case because success rates are a) very easy to calculate and b) many people think they are actually too low while I've not encountered folk saying they are too high.

We aren't talking about a system in which the other variables are unkowns or unpredictable.

[Zene]

Sorry for reviving the thread, but as a new DM this has come up several times in play already. I don’t see anything about this in the errata on a word search for “critical” —was there ever a ruling, errata, or official response?

(Personally, CF starting at -11, F starting at -1, S starting at 0, and SC starting at +10 makes the most sense to me; is easiest to calculate; and numerically seems fairest. I won’t get into the reasoning and math on that last part because it doesn’t matter and the thread is long enough. It *seems*, though, that of the two interpretations, the one I think makes less sense is the intended one. However, I’d really like to close the book on it with some kind of official word; to prevent at-table disagreements. I’d much rather have a confirmed official less-sensible way of doing it, than the possibility that either interpretation could be right depending on which part of the rules you’re reading.)

[Matthew Downie]

Based on the way the creators run it (see Jason Bulmahn references on page 5 of this discussion) the intended rule is, if DC is 12, you crit-fail on a 2 or less, and crit-succeed on a 22 or more.

I don't know if there's an official clarification anywhere, but if there was, that's what it would say.

[Squiggit]

What kind of official word are you looking for?

If you're asking whether or not Paizo's changed how critical failures work, no they haven't.

[Mathmuse]

Zene wrote:

Sorry for reviving the thread, but as a new DM this has come up several times in play already. I don’t see anything about this in the errata on a word search for “critical” —was there ever a ruling, errata, or official response?

Paizo has made no change. Nor do we expect them to change something so fundamental before Pathfinder 3rd Edition.

Zene wrote:

(Personally, CF starting at -11, F starting at -1, S starting at 0, and SC starting at +10 makes the most sense to me; is easiest to calculate; and numerically seems fairest.

Whoa, you have your math going in two directions, downward for failures and upward for successes. Clearer mathematics views each degree of success as intervals rather than endpoints. Under current rules, critical failure is the interval from negative infinity to DC-10, regular failure is the interval from DC-9 to DC-1, regular success is the interval from DC+0 to DC+9, and critical success is the interval from DC+10 to positive infinity. This is symmetrical except that the exactly-in-the-middle case of hitting the DC exactly is arbitrarily thrown into the regular success interval, which makes it slightly bigger than the regular failure interval with 10 possible rolls versus 9 possible rolls.

And subtraction by 10 is simpler to calculate than subtraction by 11.

Zene wrote:

I won’t get into the reasoning and math on that last part because it doesn’t matter and the thread is long enough. It *seems*, though, that of the two interpretations, the one I think makes less sense is the intended one. However, I’d really like to close the book on it with some kind of official word; to prevent at-table disagreements. I’d much rather have a confirmed official less-sensible way of doing it, than the possibility that either interpretation could be right depending on which part of the rules you’re reading.)

Does your reasoning and math involve information theory? I think that the maximum interest in the outcomes would be achieved at maximum information, which is when the probabilities of the four degrees of success are equal. That would be 1-5 on the d20 is a critical failure, 5-10 is a regular failure, 11-15 is a regular success, and 16-20 is a critical success. That has an entropy (total information) of 2 bits. But players will want their character's +X bonues to matter. A +1 would shift the intervals to 1-4, 5-9, 10-14, and 15-20 for an entropy of 1.98 bits and a +2 would shift the intervals to 1-3, 4-8, 9-13, and 14-20 for an entropy of 1.98. Well, the loss is trivial.

However, Paizo designers wanted crits to be rare despite information theory. The entropy of the current intervals 1, 2-10, 11-19, and 20 is 1.47 bits and of shifted interval 1, 2-9, 10-19, and 20 is 1.46 bits.

[ikarinokami]

I read this whole, thread, and I just didn't see the point of it. The language in the rule is very clear.

In PF2 you have a higher % change to crit fail than to crit succeed. and?

That's not a flaw, that's a clear intended choice, pretty clear by the unambiguous written rules.

there are some games like Shadowrun where there no critical successes and only critical failures.

If I had to guess, I would say magic the four stages of saving throws is the main reason for the asymmetrical design.

[Gisher]

Poit wrote:

And now for some numbers!

I made a spreadsheet to calculate every possible outcome for rolling a DC 20 check with modifiers ranging from +0 to +19. 20 rolls each with 20 potential outcomes - 400 combined outcomes. Here's the results:

64 critical fail (16%)
127 non-critical fail (31.75%)
144 non-critical success (36%)
65 critical success (16.25%)

So, this idea that if we interpret "fails by 10 or more" as "DC-10 or worse", we will make critical failures more likely than critical successes? I completely reject that.

If anyone would like to check the spreadsheet for errors or to make a copy to play with, here's a link. You can adjust the DC by changing cell A1. Red 1's are crit failures, yellow 2's are failures, etc.

(Note that the reason I used a DC 20 check with a scaling modifier, instead of a flat check with a scaling DC, was to prevent the rules regarding flat checks with DCs <= 1 and >= 21 from skewing the results)

This spreadsheet is nice, but it only lets you look at one case at a time. I prefer a different approach that lets me see all possible cases at once.

CRB, p. 620 wrote:

critical You can get a greater success—a critical success—by rolling 10 above your DC, or a worse failure—a critical failure—by rolling 10 lower than your DC.

So, except in the cases of a natural 20 or a natural 1, we have:

Critical Success: DC + 10 ≤ Result
Success: DC ≤ Result < DC + 10
Failure: DC – 10 < Result < DC
Critical Failure: Result ≤ DC – 10

These can be rearranged a little…

Critical Success
DC + 10 ≤ Result
DC + 10 ≤ d20 + modifier
10 ≤ d20 + (modifier – DC)

Success
DC ≤ Result < DC + 10
DC ≤ d20 + modifier < DC + 10
0 ≤ d20 + (modifier – DC) < 10

Failure
DC – 10 < Result < DC
DC – 10 < d20 + modifier < DC
-10 < d20 + (modifier – DC) < 0

Critical Failure
Result ≤ DC – 10
d20 + modifier ≤ DC – 10
d20 + (modifier – DC) ≤ -10

By defining the single parameter R to be d20 + (modifier – DC), the inequalities can be expressed in simpler forms where the results of each check can be determined by comparing R to 10, 0, and -10.

Critical Success: 10 ≤ R
Success: 0 ≤ R < 10
Failure: -10 < R < 0
Critical Failure: R ≤ -10

(Again, this excludes the cases of a natural 1 and natural 20.)

By making d20 and (modifier – DC) the axes of a grid, we can calculate R for each element simply by adding the axis values. Each column of the grid shows the possible outcomes for a die roll with a specific value for (modifier – DC). That makes it easy to mark the outcomes for each case and to calculate their probabilities.

When we adjust the top and bottom rows for the cases of a natural 20 or natural 1 we see that the distributions to the right of (modifier – DC) = 8 are all the same. Similarly the distributions to the left of (modifier – DC) = -29 are all the same. Thus there are only 40 unique cases, and we can create a universal chart of all possible checks with only 40 columns.

And here it is.
• Universal Check Tables

And to address the topic of the original post, you can easily see on the tables that the probability of a critical success is the same as the probability of a critical failure when (modifier – DC) is -10 or -11, and unequal in all other cases.

[Gisher]

ikarinokami wrote:

...

In PF2 you have a higher % change to crit fail than to crit succeed.
...

That's only true in some situations. As a counter-example, if you have a save bonus of +8 and the DC is 15 then (modifier – DC) = -7. You can see from my Universal Check Tables that in that case you would have a 20% chance of a critical success and only a 5% chance of a critical failure.

It breaks down like this:
• If (modifier – DC) ≤ -12 then critical failures are more likely than critical successes.
• If -11 ≤ (modifier – DC) ≤ -10 then critical failures are as likely as critical successes.
• If -9 ≤ (modifier – DC) then critical failures are less likely than critical successes.

[Super Zero]

Zene wrote:

Sorry for reviving the thread, but as a new DM this has come up several times in play already. I don’t see anything about this in the errata on a word search for “critical” —was there ever a ruling, errata, or official response?

There isn't anything to rule on. Success is slightly more likely because the character rolling wins ties, which generally very slightly rewards the active character.

What, are we being unfair to the concept of failure? It feels left out because it only gets nine numbers instead of ten?

ikarinokami wrote:

I read this whole, thread, and I just didn't see the point of it. The language in the rule is very clear.

In PF2 you have a higher % change to crit fail than to crit succeed. and?

That's not a flaw, that's a clear intended choice, pretty clear by the unambiguous written rules.

It's also not true.

If you have even odds of success and failure, they are equal. If you're slightly more likely to succeed, they are equal. If you're more likely than that, critical success becomes increasingly likely. Less likely, and critical failure does.