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Please see:
https://imgur.com/HnbBKs2.png
for a visual representation.
Essentially, the problem is this:
The CRB (pg 445) states:
"The rules for critical failure — sometimes called a fumble — are the same as those for a critical success, but in the other direction: if you fail a check by 10 or more, that’s a critical failure."
But that raises the question: since failing STARTS at the DC minus 1 (DC-1). Does that then mean critical failures start at DC-11, which is failure minus 10, ... or DC-10?
First glimpse, it's easy to say DC-10, but you've got to remember that the success STARTS at DC+0. My visual aid demonstrates this much better.
If it's DC-10, then you have TWICE as much of a chance of critical failure (9%) than you do of critical success (4.5%).
Regular failure happens on nine numbers, with critical failure happening on the 10th.
Regular success happens on ten numbers, and critical success happening on the 11th.
If it's DC-11, then it's all equal.
Regular failure on 10 numbers, critical failure on the 11th.
Regular success on 10 numbers, critical success on the 11th.
In an edition where simplicity is the goal, part of me wants to err on the side of simplicity and say critical failure is just DC-10... but I know the rules lawyers love these games, too. The math nerd in me hates this, as well - especially when +1 or -1 matters in everything else we do.
A typical fencepost issue. Do we start counting from zero or from one.
So, the first failure is at DC-1, since if we make the DC, we have a success. Then, counting down, we have:
DC Success
DC-1 Failure by one.
DC-2, Failure by two.
...
DC-10, Failure by ten (or more). Equals critical failure.
So, RAW, Critical failure is at DC-10. I'm not sure if that is RAI.
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if you fail a check by 10 or more, that’s a critical failure.
If the DC is 20 and you rolled a 19, how much did you fail the check by?
If you rolled an 18, how much did you fail the check by?...
If you rolled a 10?
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I'm not sure if that is RAI.
I assume it's intentional so that the math is simply +10 and -10 rather than one of them being either 9 or 11 to keep the probabilities nice
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I think it is DC-11.
In order to Succeed, you have to at least match the DC (15 on your chart). To Critically Succeed, you must beat the DC by 10 or more (So a 25 on your chart).
Similarly, to fail a check, you must roll 1 or more less than the DC (so a 14 on your chart) To Critically Fail, you have to roll 10 or more less than what you needed to Fail (so a 4 on your chart)
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So, the first failure is at DC-1, since if we make the DC, we have a success. Then, counting down, we have:DC Success
DC-1 Failure by one.
DC-2, Failure by two.
...
DC-10, Failure by ten (or more). Equals critical failure.
If we did that on the other side, then
DC+0 = Success by one.
DC+1 = Success by two.
DC+2 = Success by three.
...
DC+9 = Success by ten, critical success.
i.e.: 15 DC would critically succeed on a 24, not a 25, using that same reasoning.
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This is a problem that had not escaped my notice, being the math nerd that I am (although I imagine we'll be seeing Mathmuse in this thread shortly, leaving me quite outclassed :P ).
I suspect that the ease of counting is going to have more of a positive effect on play than the slight inaccuracy of the math will have a negative effect, though.
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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. 445–446 critical hit (Strike) 471 critical specialization (weapons) 283–284
While you can argue about what the critical fail number should be, I don't think there's really a grey area about what it is, by written rules.
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It's DC-11, because hitting the DC is a success. This makes it slightly less easy to calculate on the fly, but now that we all know this it's easy enough to point out "no, a 15 isn't a crit fail on a DC25, but a 14 is".
You guys are confusing me. lets say the DC is 10 my assumption would be a 0 is a crit fail and a 20 is a crit success. Am I thinking about this wrong?
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To have the odds be even for a critical success and a critical failure, a -1 would be your critical failure.
To use the rules that got published, a 0 is a critical failure.
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At DC+0 = success
DC+1 = success + 1Failure is measured by how many more points would you have needed to succeed. Critical success is measure by how many points did you have above and beyond what you needed. This is how I read the rules when I was going through it.
But then the mirror of that would also be true.
i.e.:
DC-1 = failure
DC-2 = failure -1
DC-3 = failure -2
...
DC-11 = failure -10
So at a 15 DC, failure is from 5 to 14, and critical failure at 4 and below. No matter how you cut it mathematically, there isn't a "zero" to start from; you have to be positive or negative in your number line.
I just hope Paizo will clarify their intent.
I'm fine with either answer, I see good reasoning for both points. I just want to know: are we officially going for simplicity, or are we going for even odds?
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So, the first failure is at DC-1, since if we make the DC, we have a success. Then, counting down, we have:DC Success
DC-1 Failure by one.
DC-2, Failure by two.
...
DC-10, Failure by ten (or more). Equals critical failure.
If we did that on the other side, then
DC+0 = Success by one.
DC+1 = Success by two.
DC+2 = Success by three.
...
DC+9 = Success by ten, critical success.i.e.: 15 DC would critically succeed on a 24, not a 25, using that same reasoning.
Except the wording is "beating the DC by 10" and I interpret "beating the DC" by 1 to be one higher than success not just equal to the DC.
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Except the wording is "beating the DC by 10"
No it isn't.
"You critically succeed at a check when a check’s result
meets or exceeds the DC by 10 or more. If the check is an
attack roll, this is sometimes called a critical hit. You can
also critically fail a check. The rules for critical failure—
sometimes called a fumble—are the same as those for a
critical success, but in the other direction: if you fail a
check by 10 or more, that’s a critical failure" Page 445, CRB
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DC=X.
Critical success: check result ≥ X+10
Critical failure: check result ≤ X-10
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For those trying to wrap their heads around the number scale problem, I tried to clarify a bit better here.
https://i.imgur.com/dXEIEtX.png
Pay special attention to the Success and Failure + or - in comparison to the DC + or -
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DC=X.
Critical success: check result ≥ X+10
Critical failure: check result ≤ X-10
Yes, maybe, but you see the problem, don't you?
Success is also the DC, meaning DC and 10 more, 11 total places on a number scale.
In the method you describe, means 10 less and minus the DC itself, 9 total places on a number scale. You hit critical failures sooner than you hit critical successes.
https://i.imgur.com/dXEIEtX.png
This is why it's a conundrum, and I really hope Paizo answers this. One method or the other, there is a loss: a loss of simplicity, or a loss of equal statistical outcomes.
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Personally I am fine with the loss of equal statistical outcome. 'Twice as likely' is a very nice soundbyte, but we are talking about a maximum absolute probability change of 5%.
The ease of calculating is much more important to me. And the young kids that I play with who are only starting to learn math.
Except the wording is "beating the DC by 10"No it isn't.
"You critically succeed at a check when a check’s result
meets or exceeds the DC by 10 or more. If the check is an
attack roll, this is sometimes called a critical hit. You can
also critically fail a check. The rules for critical failure—
sometimes called a fumble—are the same as those for a
critical success, but in the other direction: if you fail a
check by 10 or more, that’s a critical failure" Page 445, CRB
"Beating the DC by 10 or more is referred to as a critical success" Pg 10
Good job Paizo
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And how did you come up with 4.5% vs. 9% anyway? d20 has a very nice 5% per face calculation. You don't have to be Mathmuse to calculate things. Especially since opposed rolls virtually all went away.
Ah. I see. Your chart has 22 numbers on it.
That is not possible to be rolled on a d20. There is no actual DC that is going to result in chances of getting both a crit success and a crit fail based just on the DC +/-10.
If 11 is the DC, then a 1 would be a crit fail, but you would need a 21 in order to crit succeed. If the DC is 10, then a 20 would crit succeed but you would have to roll a 0 on the die in order to crit fail.
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Yes, but you see the problem, don't you?
No. I do not. Critical Failures are the same as critical successes, just in the other direction.
Succeeding by ten or more means ten spaces or more to the right of the DC on the number line. A check result of 25 for a DC of 15.
Failing by ten or more means ten spaces or more to the left of the DC on the number line. A check result of 5 for a DC of 15.
If you are going to go by your own graphic then you can't say DC+10 is a pretty blue critical success, but a DC-10 isn't a dangerous red critical failure.
Either you are going to have Red and Blue start at DC ± 10 or Red and Blue starts at DC ± 11.
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That's the whole purpose of all of this: what's the reference point when starting to determine failure?
Critical Success is the DC+10, the reference point is the DC, where success starts, meaning Critical Success begins on the 11th point of the Success scale.
Inversely, Critical Failure is "failing a check by 10 or more," but do we start with Failure as a reference point? That technically begins at DC-1, not at DC-0.
Put it this way: Critical Success begins on the 11th point of the Success scale. Wouldn't Critical Failure also (because it's "the same") begin on the 11th point of the Failure scale, then?
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"Beating the DC by 10 or more is referred to as a critical success" Pg 10
Okay, granted. The brief summary in the introduction does have that phrasing.
However, it is still the same difference. Literally, since difference is exactly what is being determined here.
16 beats 15 by one is the same as 16-15=1.
25 beats 15 by ten is the same as 25-15=10.
14 fails 15 by one is the same as 15-14=1.
05 fails 15 by ten is the same as 15-5=10.
Fifteen beats fifteen by ZERO. 15-15=0
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And how did you come up with 4.5% vs. 9% anyway? d20 has a very nice 5% per face calculation. You don't have to be Mathmuse to calculate things. Especially since opposed rolls virtually all went away.
Ah. I see. Your chart has 22 numbers on it.
That is not possible to be rolled on a d20. There is no actual DC that is going to result in chances of getting both a crit success and a crit fail based just on the DC +/-10.
If 11 is the DC, then a 1 would be a crit fail, but you would need a 21 in order to crit succeed. If the DC is 10, then a 20 would crit succeed but you would have to roll a 0 on the die in order to crit fail.
Yeah, that's what I meant when I say there's no "zero" in this. The DC is on the "positive" side of the scale; there has to be a negative counterpart/reference-point. Both sides of the equation need balanced for the scale to be equal.
It's kind of like looking at a 5' grid. You can't be on the line, you have to be in a square (Medium creatures, assumed).
If you start measuring inside the house at the first 5' square (with the door on the threshold line), then +10' inside is just 2 squares from the door.
Just because you're standing in the first square, doesn't mean that 10' outside the house is the first square outside, because 10' inside is only 1 square away. No, outside begins at square 1 as the first frame of reference, and is 2 squares from the door.
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That's the whole purpose of all of this: what's the reference point when starting to determine failure?
DC-1 through DC-9 is an ordinary failure. DC-10 or more is a critical failure.
DC+0 through DC+9 is an ordinary success. DC+10 or more is a critical success.
Edit- Fixed... it is almost midnight and I am tired. DC +-0 meets the DC. DC+1 exceeds it. Both are successes.
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It's kind of like looking at a 5' grid. You can't be on the line, you have to be in a square (Medium creatures, assumed).
If you are looking at a 5' grid, then the space you are starting in is zero. A five foot step is one square over. A 50' Stride is ten squares over.
You literally made a number line with a zero point and at the same time are saying "there is no zero."
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It's kind of like looking at a 5' grid. You can't be on the line, you have to be in a square (Medium creatures, assumed).If you are looking at a 5' grid, then the space you are starting in is zero. A five foot step is one square over. A 50' Stride is ten squares over.
You literally made a number line with a zero point and at the same time are saying "there is no zero."
No, sorry. See this as for why. The reference point isn't a middle point, the reference point of a DC is already on one side of the scale.
https://i.imgur.com/VH3taTN.png
You are conflating the reference point and the center of the scale, which is simply not true.
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The center of the scale is not a whole number (an integer). It cannot be. You must start on an integer, which means that an equal number of points must be on either end of the scale.
Just like the center for any normal, even-sided dice is not a number that can be rolled.
The middle of a d20 is 10.5, it is not an integer. But because 20 is an even number and you start on 1 (not 0), then the lower half is 1 through 10, the higher half is 11 through 20. That means each half has ten numbers, or ten points on each side of the scale.
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The problem with a DC is that meeting it is a success, meaning it puts it on one side of the scale. It is not the fulcrum of the scale, it is the first weight placed on one side.
Literally, the reference point (DC) is the first success point on the outcome scale.
To make an equal counterpart for the scale to balance, there must be a reference point as the first failure point on the scale, which is DC-1.
S1 = DC+0 and F1 = DC-1, right? Then it can only follow...
S2 = DC+1 and F2 = DC-2
S3 = DC+2 and F3 = DC-3
...
S11 = DC+10 and F11 = DC-11
That's the only way the scale doesn't tip in one direction or the other. 11 points on both sides. 22 points/numbers/places total. That's it.
DC-10 is not equal in scale to DC+10. It can't be.
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I think I get it now.
That's the only way the scale doesn't tip in one direction or the other.
Seems to me that for things to be truly balanced, meeting the DC exactly shouldn't be a success or a failure, it should be a neutral result...
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You critically succeed at a check when a check’s result meets or exceeds the DC by 10 or more.
What is the word "meets" doing in that sentence? How can you meet the DC by 10?
You critically succeed at a check when a check’s result meets or exceeds the DC by 10 or more.What is the word "meets" doing in that sentence? How can you meet the DC by 10?
Presumably just to clarify that the tipping point is DC+10. If it was only 'exceeds' I think you might have some people arguing that it needs to be DC+11 to crit.
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Zero is an integer. This is very important.
Here is a number line. It has integers on it.
Here is your number line. You might see some similarities.
If you roll a 20 on your chart, that is a margin of +5. If you roll a 15 on the same chart that is a margin of 0. If you roll a 10, that is a margin of -5.
By the chart that you made, let the check result be y and the DC be x. Let x=15.
20-15=+5
15-15= 0
10-15=-5
As the book says, "if your result is equal to or greater than the DC, you succeed!"
Anything where y-x is zero or more is a success. A positive margin of 10 or more is also a critical success.
y-15=10
Anything where y-x is -1 or less is a failure. A negative margin of 10 or more is also a critical failure.
y-15=-10
Solve for y.
I'll give you a hint. y is 25 in the first and 5 in the second.
tldr version?
DC=X.
Critical success: check result ≥ X+10
Critical failure: check result ≤ X-10
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Zero is an integer.
Obviously, but we're not measuring "nothing" in this application of number lines - we're measuring EITHER/OR, POSITIVE or NEGATIVE, that is, FAILURE or SUCCESS. There cannot be a "middle" with no value.
Your starting point is on the SUCCESS side. A DC is a success.
Therefore, a DC must be on ONE SIDE of the scale, NOT the middle.
Your own graph states clearly, zero is neither negative nor positive. A total of a DC is a success, not nothing.
If you absolutely must place a zero, you put it on the line between the two sides and you can't actually reach the number; It is the fulcrum of the scales. It only occurs when the two sides are balanced.
EDIT: It's like the d20 rolls we mention. "Zero" as a middle point (10.5) theoretically exists as a division between measurement categories, but it cannot happen in reality. You will always end up rolling a low-end number (1-10, a failure on the scale), or a high-end number (11-20, a success on the scale). Negative or positive. Failure or success. There is no "null," a "nothing" roll is not possible.
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There will never be a roll that can both critically fail and critically succeed without the 1 and 20 special rules.
As such this is a bunch of pitpickery that doesn't matter.
It's better to be simple and fast to use "DC+10, DC-10" than it is to be mathematically "perfect" whatever that means.
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https://i.imgur.com/dXEIEtX.png
Your number line illustrating this problem contains 22 numbers. We are not rolling 22-sided dice. This is not a problem for 20-sided dice.
Also note that natural 20s and 1s adjust the degree of success up or down. Unless the roll is trivially easy or exceedingly hard, you will always have at least a 5% chance to critically succeed and at least a 5% chance to critically fail.
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There cannot be a "middle" with no value.
You are absolutely right.
There is no "middle" with no value. There is only failure and success. Either/or. Nothing in between.
Your mistake is in saying that failure is negative and success is positive. This is not true.
The equation remains "(check result)-(difficulty class)." Success is every check result that is the same value or is greater than the DC. That means y-x can be 0. Zero is the threshold for success.
That leaves us with only two options.
Failure is in the set {..., -3,-2,-1}
Success is in the set {0,1,2,3,...}
No middle ground between the two.
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https://i.imgur.com/dXEIEtX.pngYour number line illustrating this problem contains 22 numbers. We are not rolling 22-sided dice. This is not a problem for 20-sided dice.
Because this isn't about d20s alone, this is about the spectrum of results that can mean one of four possibilities: critical failure, failure, success, and critical success. The absolute minimum number of points on a scale to represent this is 22 points.
Because we're using DC+10 as the high end of the spectrum, that means the DC (a number) + ten more numbers. That's 11 points on the positive end of the scale. The negative side of the scale needs an equal number of points. 11 + 11 = 22.
But your reference of a d20 does have a point: as NA Palm put it, you cannot have a DC where it is possible for you to critically succeed and fail, not without the special nat-1 and nat-20 rules.
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https://i.imgur.com/dXEIEtX.pngYour number line illustrating this problem contains 22 numbers. We are not rolling 22-sided dice. This is not a problem for 20-sided dice.
Also note that natural 20s and 1s adjust the degree of success up or down. Unless the roll is trivially easy or exceedingly hard, you will always have at least a 5% chance to critically succeed and at least a 5% chance to critically fail.
That's beside the point. rolling 1 or 20 is only changing the outcome after the degree of success has been calculated.
The point being argued about is:
If DC15 is a success, a result of 25 is a critical success. Is a result of 5 a failure or a critical failure?
There are 10 results that indicate a success (before 1/20): 15, 16, 17, 18, 19, 20, 21, 22, 23, 24.
To get 10 (not-critical) failure results (before 1/20), these need to be: 14, 13, 12, 11, 10, 9, 8, 7, 6, 5. This 5 is being contested.
-----------------------------
Edit:
Determine the Degree of Success
You critically succeed at a check when a check’s result meets or exceeds the DC by 10 or more. If the check is an attack roll, this is sometimes called a critical hit. You can also critically fail a check. The rules for critical failure—sometimes called a fumble—are the same as those for a critical success, but in the other direction: if you fail a check by 10 or more, that’s a critical failure.
If you rolled a 20 on the die (a “natural 20”), your result is one degree of success better than it would be by numbers alone. If you roll a 1 on the d20 (a “natural 1”), your result is one degree worse. This means that a natural 20 usually results in a critical success and natural 1 usually results in a critical failure. However, if you were going up against a very high DC, you might get only a success with a natural 20, or even a failure if 20 plus your total modifier is 10 or more below the DC.
That last part there, should make it clear. For a DC of 15, getting a result of 5 is a failure, but not a critical failure (before applying the 1/20 adjustment).
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There cannot be a "middle" with no value.Your mistake is in saying that failure is negative and success is positive. This is not true.
The equation remains "(check result)-(difficulty class)." Success is every check result that is the same value or is greater than the DC. That means y-x can be 0. Zero is the threshold for success.
That leaves us with only two options.
Failure is in the set {..., -3,-2,-1}
Success is in the set {0,1,2,3,...}
No middle ground between the two.
You're conflating several things here.
Yes, the number zero can obviously exist on the number line, like so:
https://i.imgur.com/Q1HLs0I.png
That is not what I'm talking about here. You are treating the DC as if it were a zero in the sense of a NULL. It's not. It is a positive result, from the standpoint of a success vs. a failure. Again, my graphic chart illustrates this perfectly, where DC=0 but it also equals Success 1. Thus DC = DC+0 = S1. These things can all be true at the same time, I really don't understand why you chose to split hairs here on semantics when my points are clear.
So if our reference point, the DC, or DC+0, or S1 is equal to a SUCCESS, which is a positive result, then it follows that DC+10 is 11 total points on a the number scale. The reference point is POSITIVE, not NULL. Period.
Where it gets wonky is that the failure scale begins at DC-1, or F1. A negative result; a FAILURE.
The question about this entire thread is whether or not the DC of a critical success is measured as DC-10, or F-10 (which is DC-11). The former method is easier, but it is not even-scaled, with only 10 points on the Failure scale, vs 11 points on the Success scale.
You even got the set portions of this all correct, but you're skipping places:
In my first example (DC15):
Success = {15, 16, 17, 18, 19, 20, 21, 22, 23, 24} with 25+ C.S.
Failure = {14, 13, 12, 11, 10, 09, 08, 07, 06, 05} with 04- C.F.
In my second example (DC0):
Success = {00, 01, 02, 03, 04, 05, 06, 07, 08, 09} with 10... C.S.
Failure = {-01, -02, -03, -04, -05, -06, -07, -08, -09, -10} with -11... C.F.
(the font type made it not line up, I tried to ease that a bit with leading zeros)
Well I think most of this discussion is centering about the beat by / fail by definitions.
If the DC is 15, then in my book beat by 1 would be if you rolled a total of 16 (because this is 1 higher than required) and failed by 1 would be if you rolled a total of 14 (because this is 1 less than required).
Going from here beat by 10 or more is 25+ and failed by 10 is 5- , which by the way is the same as if you are using DC+10 or DC-10.
15 + 10 = 25
15 -10 = 5
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Here's another way to put it:
https://i.imgur.com/JW5xHDD.png
Instead of "positive result" think "right side"
Instead of "negative result" think "left side"
No matter how you cut it, if your starting point is a finger (an integer), then the distance from the far side of that hand to that reference finger will never, ever be equal to the distance from the opposite hand's far side to the reference finger.
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https://i.imgur.com/dXEIEtX.pngYour number line illustrating this problem contains 22 numbers. We are not rolling 22-sided dice. This is not a problem for 20-sided dice.
Also note that natural 20s and 1s adjust the degree of success up or down. Unless the roll is trivially easy or exceedingly hard, you will always have at least a 5% chance to critically succeed and at least a 5% chance to critically fail.
That's beside the point. rolling 1 or 20 is only changing the outcome after the degree of success has been calculated.
The point being argued about is:
If DC15 is a success, a result of 25 is a critical success. Is a result of 5 a failure or a critical failure?
There are 10 results that indicate a success (before 1/20): 15, 16, 17, 18, 19, 20, 21, 22, 23, 24.
To get 10 (not-critical) failure results (before 1/20), these need to be: 14, 13, 12, 11, 10, 9, 8, 7, 6, 5. This 5 is being contested.
If the DC is 15, then...
A result of 14 fails by 1.A result of 13 fails by 2.
A result of 12 fails by 3.
A result of 11 fails by 4.
A result of 10 fails by 5.
A result of 9 fails by 6.
A result of 8 fails by 7.
A result of 7 fails by 8.
A result of 6 fails by 9.
A result of 5 fails by 10.
And as quoted in the initial post in this thread:
The CRB (pg 445) states:
"The rules for critical failure — sometimes called a fumble — are the same as those for a critical success, but in the other direction: if you fail a check by 10 or more, that’s a critical failure."
Failing by 10 or more results in a critical failure. The 5 is a critical failure.
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Well I think most of this discussion is centering about the beat by / fail by definitions.
That is indeed the crux of it: the book doesn't state explicitly that it is DC-10, only, "If you fail a check by 10 or more, that’s a critical failure"
But if failing starts at DC-1... do we count from the start of failing, or do we count from the start of succeeding?
One method is simpler, but unequal. The other is equal, but can cause a brief delay/complication. I see good reasons for both solutions.
Failing by 10 or more results in a critical failure. The 5 is a critical failure.
But this is never stated explicitly, you assume intent to mean one way. This is why it is an issue; the language isn't clear.
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Well I think most of this discussion is centering about the beat by / fail by definitions.That is indeed the crux of it: the book doesn't state explicitly that it is DC-10, only, "If you fail a check by 10 or more, that’s a critical failure"
But if failing starts at DC-1... do we count from the start of failing, or do we count from the start of succeeding?
One method is simpler, but unequal. The other is equal, but can give pause. I see good reasons for both solutions.
The book does state this explicitly. HammerJack cited it earlier.
While you can argue about what the critical fail number should be, I don't think there's really a grey area about what it is, by written rules.
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. 445–446 critical hit (Strike) 471 critical specialization (weapons) 283–284
5 is 10 lower than 15. A result of 5 on a DC 15 check is a critical failure.