Traits of Exotic d20 Substitutes pt 3: The Really Weird
Lots of die configurations can substitute for a d20, or for 3d6. This article looks at some of the most unusual. Part 3 of 3.

The image of the balance is by Anna Varsányi from Pixabay. I’ve changed it’s balance, added a load of dice, and changed the background color.

I made the time-out logo from two images in combination: The relaxing man photo is by Frauke Riether and the clock face (which was used as inspiration for the text rendering) Image was provided by OpenClipart-Vectors, both sourced from Pixabay.
There’s something indescribably appropriate about writing the first words of this post on Halloween – after all, many of these rolls are monsters unfit for gentle company. At the same time, some of them might get under your skin and make themselves at home, because there are some absolutely fascinating (not to mention strange) alternatives being put under the microscope today!
Because the die rolls are so strange, I’ve decided that each graph will be linked to a larger version that can be opened in a separate tab by clicking on the thumbnail. I’m also toying with the notion of doing some even larger versions in a PDF – if so, I’ll feature the link to it prominently.
I’m kicking things off today with a last-minute extra inclusion just as a warm-up. Although conceptually wild, it’s by far the tamest alternative on show today!
BONUS EXTRA: Exotic Choice #0a: 2d6+1 (for high results desired) or 2d6+6 (for low results desired)
I came up with this while finalizing the formatting of the previous post; when a couple of the things I had written about caught my eye in succession and sparked new thoughts.
Specifically: what if the roll was 3d6 – but one of the dice was fixed, in the opposite direction of what a character wants to roll to succeed? A ‘1’ if they want to roll high, a ‘6’ if they want to roll low?
In form, this would then become a triangular probability curve, because it’s functionally the same as 2d6 plus modifier – against a target intended for 3d6. That modifier is critical – the average roll of a d6 is 3.5, so a 1 effectively means a -2.5 modifier against a target intended for 3d6 when you are trying to roll high, and a 6 means a +2.5 modifier on 3d6 when you are trying to roll low.
Integer values matter when they trigger a binary choice like that. In the Hero System, several defined rolls set the standards: 5/-, 7/-, 11/-, 14/-, and 17/-. These are all attempting to roll low, to get below the target number. In D&D, back when it was still 3d6 based, you often had to roll high but sometimes you had to roll low – it depends on what you’re rolling for. With 3rd Ed, this was cleaned up so that you were always trying to roll higher than the target. So both variations have to be evaluated. To do so, I’ll use the same standards – but look at rolling 5+, 7+, 11+, 14+, and 17+ – even though that edition also shifted to the d20. So this is a legitimate option for replacement of both.
With the Hero System rolls, the higher the target number, the easier the roll is supposed to be. With ‘modern’ D&D and Pathfinder, the higher the target number, the harder the roll.
We start, as usual, with some probability graphs:

Every result in between the two curves is bad news for the rolling character. Click the thumbnail for 1024 x 361 version.
Min, Max, Ave
2d6+1:
Minimum 3
Maximum 13
Average 8
26+6:
Minimum 8
Maximum 18
Average 13
The fact that one peaks as the other begins makes me kinda curious about what the sum of the two – 4d6+7 – would look like, but that’s outside the scope of this article.
The Thresholds
The 1% Threshold
Everything beats this minimum – no valid results are off the table.
The 3% Threshold
On 2d6+1, 3 and 13 are just below this threshold. On 2d6+6, 8 and 18 are in the same category. In both cases, it’s the most extreme results only; everything else is in the next threshold group or higher.
In fact, there’s nothing in the 3%-5% band, either. The probability is rising too quickly for that.
The 5% Threshold
Breaking the 5% threshold but not making it to the next, 10% mark, are a couple of results on each side of each of the curves.
2d6+1: 4-5 and 11-12; 2d6+6: 9-10 and 16-17. So these results are more likely to come up than on a d20.
The 10% Threshold
Between 10% and 15% are also two results from each side of the curve.
2d6+1: 6-7 and 9-10. 2d6+6: 11-12 and 14-15. These results are more likely to come up than on a d10.
The 15% Threshold
That leaves only the absolute peaks of both ‘curves’, 8 and 13 respectively. They aren’t much higher than 15% but they legitimately beat that target. In fact, these results have the same probability as a flat 1d6 roll plus modifiers.
Slices Of Range: Percentages Of Probability
Range Of Results
3-13 and 8-18 have exactly the same range of results, which is not all that surprising since they are both 2d6 rolls. 11 results in each. The odd number means that there is a single result that represents the peak probability – until you get into the exotic die rolls to come, anyway!
Ave – Min, Max – Ave
-
These values will also be the same in all four cases – 8-3=5, 13-8=5, and 18-13=5.
1/3 (Ave-Min) + Min
Here’s where things have to diverge because the two rolls have different minimum values.
1/3 of 5 is 1.6667, which will be common to both.
1.6667 + 3 = 4.6667, so 3 & 4 are the lowest tier of results for 2d6+1. They have a combined probability of 8.33%.
1.6667 + 8 = 9.6667, so 8 & 9 are the equivalents (with the same combined probability) for 2d6+6.
2/3 (Ave-Min) + Min
2/3 of 5 is 3.3333, again common to both because it’s a function of the 2d6 part of the rolls.
3.3333 + 3 = 6.3333, so 5 and 6 are the middle lower results band for 2d6+1. They have a combines probability of 27.78 – 8.33 = 19.45%.
3.3333 + 8 = 11.3333, so 10 and 11 are the equivalents for 2d6+6, with the same probability.
The Lower Core
That means that 7 and half of 8 comprise the lower core for 2d6+1 – that’s 13.89 + 1/2 x 16.67 = 22.225%.
The 2d6+6 equivalents, with the same probability, are 12 and half of 13.
The Upper Core: 1/3 (Max-Ave) + Ave
Starting on the downhill leg of the probability charts, we have another 22.225% representing 9 and the other half of 8 on 2d6+1, and 14 and the other half of 13 on 2d6+6.
2/3 (Max-Ave) + Ave
Those are followed by the upper middle, a combined probability of 19.45% again, and a span of 2. For 2d6+1, that’s 10 & 11, and for 2d6+6, it’s 15 & 16.
The Lofty Outcomes
The very best results, with a probability of 8.33%, are 12 & 13 on 2d6+1, and 17 & 18 on 2d6+6.
2d6+1:
03-04: 8.33%
05-06: 19.45%
07-08: 22.225%
08-09: 22.225%
10-11: 19.45%
12-13: 8.33%
2d6+6:
08-09: 8.33%
10-11: 19.45%
12-13: 22.225%
13-14: 22.225%
15-16: 19.45%
17-18: 8.33%
No real surprises in this set of results except possibly the closeness of 19.45% to 22.225% – especially given the threshold indicator that the probability slope is quite steep with 2d6.
Slices Of Probability: The Definitive Result Values
Slicing up the 100% pie into 5 slices as equally as possible is the name of the game in this subsection.
The Lowest 20%
20% falls after the third result on each curve, so the lowest 20% of results comprise outcomes of (2d6+1) 3-5 and (2d6+6) 8-10. I think it’s just a coincidence that the upper limit of one is double the upper limit of the other.
Second Lowest 20%
21-40% contains only a single result in each case – 6 for 2d6+1 and 11 for 2d6+6.
The Middle 20%
41-60% contains two values, including the peak. For 2d6+1, those are 7-8, and for 2d6+6, 12-13.
Second-Highest 20%
61-80% again holds just one result – 9 for 2d6+1, and 14 for 2d6+6.
Highest 20%
Which means the highest 20% of rolls will contain the results from 10-13 for 2d6+1 and 15-18 for 2d6+6.
Peak Probability
In both cases the peak probability is 16.67%.
Matching Result: 1/3 Peak Probability
1/3 x 16.67 = 5.5567%. This lands in between 3 & 4 (and 12 & 13) on 2d6+1, and between 8 & 9 and 17 & 18 on 2d6+6. So, once again, only the most extreme results are chosen by this method. That’s actually rather predictable, given the earlier threshold results, since 5.5567 is so close to the 5% threshold.
Matching Result: 2/3 Peak Probability
2/3 x 16.67 = 11.1133%. As it happens, there are results that have 11.11% probability of occurring, and so these would have to be right on this line. On 2d6+1, these are 6 and 10 – so 4-6 are in this probability zone, as are 10-12. The 2d6+6 equivalents are, predictably, 5 higher – 9-11 and 15-17.
The most probable results are therefore 7-9 (on 2d6+1) and 12-14 (on 2d6+6).
2d6+1:
01-20%: 3-5 (span 3)
21-40%: 6 (span 1)
41-60%: 7-8 (span 2)
61-80%: 9 (span 1)
81-100%: 10-13 (span 4)
< 1/3 peak probability: 3 (span 1)
1/3 – 2/3 peak probability: 4-6 (span 3)
2/3 – peak – 2/3 peak: 7-9 (span 3)
2/3 – 1/3 peak probability: 10-12 (span 3)
< 1/3 peak probability: 13 (span 1)
It’s the evenness of the spans in the latter table that are most telling. While there is clearly a peak probability associated with the innermost results, there is a significant chance of a result outside them. In fact, there is a 100 – 13.89 x 2 – 16.67 = 55.55% chance that the result of any given roll will be outside the 7-8-9 peak.
2d6+6:
01-20%: 8-10 (span 3)
21-40%: 11 (span 1)
41-60%: 12-13 (span 2)
61-80%: 14 (span 1)
81-100%: 15-18 (span 4)
< 1/3 peak probability: 8 (span 1)
1/3 – 2/3 peak probability: 9-11 (span 3)
2/3 – peak – 2/3 peak: 12-14 (span 3)
2/3 – 1/3 peak probability: 15-17 (span 3)
< 1/3 peak probability: 18 (span 1)
And these are exactly the same, just 5 higher on the results.
Summary Of Results
The bottom line in terms of mechanics is that you are taking a d6 away from the character’s roll and replacing it with the worst possible outcome.
But I also have to make the point that you can work it in the other direction – choosing the option that is most beneficial to a character’s chances of success.
When To Use This Substitute
That matters because of what this die roll is saying to whoever runs that character. If it’s the more difficult option, you are telling the operator of the character, “I want this roll to fail and I want to be sure that you know that”. Or, more simply, “This roll deserves to fail.”
The alternative construction, that benefits the character’s chances of success, says, “I want this roll to succeed and I don’t care who thinks I’m being biased.”
In other words, this construction should be reserved for those occasions when the whole point of the roll is making that statement. When a move is so brain-dead stupid that it doesn’t deserve even the minimal chance of success it might have on 3d6 or d20.
So I guess I need to actually compare what the chances of success are for different targets.
Target 17/- (17 or less)
With 3d6, you have a 99.54% chance of making this target.
With a d20, it’s 85% chance.
With the penalizing construction (2d6+6), it’s 97.22%.
With the advantageous construction for an “or less” roll (2d6+1), it’s 100% certain that you will succeed.
Target 14/-
With 3d6, your chances of success are 90.74%. With a d20, it’s 70%.
With 2d6+6, it’s 72.2% – so better than on a d20, but not by much. Compared to a 3d6 roll, you are way worse off.
With 2d6+1, it’s still 100% success.
Target 11/-
3d6 gives a 62.5% chance of success. A d20 gives 55%.
2d6+6 gives 27.78% chance. That’s like half the chance of a d20.
And, for the first time, not even 2d6+1 makes success certain – there is a 91.67% chance of success, so the odds are way better than ‘normal’.
Target 8/-
3d6 has only a 25.93% chance of making this roll. 3 times in 4, roughly, you would expect to fail. On a d20, the chances are a little better at 40%, but the odds are still stacked against you a little.
2d6+6 has just a 2.78% chance of success. It literally takes the lowest roll possible to make this target. If both the dice aren’t snake eyes, you’ve failed.
2d6+1 has a better shot at it – 58.33% – but you’re still going to fail almost half the time. This is actually a fairly hard target to achieve!
Target 5/-
…but not as hard as this target. On 3d6 you have just a 4.63% chance. On a d20 it’s a little over 5 times that, at 25%.
2d6+6 – forget it, your lowest result is an 8, so a 5 or less is not an option.
Even the construction that appears to give as good a chance at success as you are likely to get, a 2d6+1, has only a 16.67% of success – so a d20 is actually the more generous option with a target this low.
It’s much the same story if you look at rolling X or more, just in the other direction. The 2d6+6 becomes the generous option, and the 2d6+1, the handicapping one.
Either way, this choice is all about the message; the actual die roll is almost superfluous.
Exotic Choice #8: d4 x d6 – d4 +1 or +4
Now, things start getting strange. For this, you need two different colored d4s and a d6. One d4 is designated the multiplier; whatever shows on the face of that die gets multiplied by whatever’s showing on the d6. The usual nomenclature around me borrows from the d% – the multiplying d4 is “high”.
The +1 option is for replacing a 3d6 roll, the +4 is for replacing a d20 roll.
Originally, I had this listed with no modifier whatsoever, but I was looking at the resulting probability chart and thinking about the prospects of replacing d20 and 3d6, and the modifiers suddenly made a lot of sense to me.
Let me explain why. a d20 has results from 1 to 20, yes? The native construction of this roll gave results from -3 to 23. Which puts the mid-point of the results (NOT the average!) at 10. Bu the bulk of the probability is below this, at around 0-5. A +4 modifier shifted the curve to the right, because that’s what positive modifiers do – the middle of the range becomes 14, and the average will be 4-9. That makes it a usable substitute, if one that’s heavily weighted low.
3d6 ranges from 3-18. The significant probability results of the native curve end at around 11. So adding 6 shifts the minimum to a 3d6-comparable 3, the middle of the range to about 16, the peak probability to 6-11, and the end of that significant results range up to about 17, again making this a usable substitute for the 3d6 roll, again one that is biased low.
By applying the different modifiers, it makes both versions fit for purpose and the advice regarding the use of this construction, the same, or close enough to it.
With that addressed, let’s talk about the core of the roll. Multiplied die rolls have a singular characteristic: they bulk the probabilities low, but have long tails leading off into higher values. These come at a penalty – certain results that simply can’t happen. There’s no multiple that leads to a result of 17, for example – it’s a prime number.
To solve this issue, you either have to add a die roll or subtract one. Adding one extends the length of the tail by the size of the added die roll, subtracting one shortens it. Adding one also shifts the probabilities right by the average of the added die, while subtracting shifts it left by that average.
Once you’ve decided to use a multiplied die roll, you’re then negotiating a compromise between the native result and a useful configuration by way of the added or subtracted die roll. The smaller you make it, the smaller the impact – so I thought hard about d2 and d3, but decided that d4 was small enough in this instance. I also considered d5 and d6, but thought that the impact of the larger die was too significant. So that’s why this offering is d4 x d6 – d4 + 4 or 6.
I’m going to introduce a new way of writing die rolls, having typed that sequence once too often for it to be convenient.
It’s a simple extension of what’s already done – low dice size to high within an expression, ending with ‘d0’ i.e. modifiers. The new part is a way to indicate Conditional Changes.
In this case,
d4 x d6 – d4 +4,6 [d20r,3d6r]
The conditional parts are separated by a comma instead of text and are followed by a symbolic representation of the condition for differentiation between the two. Once that has been established, in whatever context you are using this notation, you can leave off the content of the square brackets, with the empty brackets meaning “as before”:
d4 x d6 – d4 +4,6 [ ]
So, for example, you might have the following as a legitimate construct for some purpose:
d4,d6 [a,b] x d6 – d6 +1,10 [ab,c]
[a] = d20r
[b] = 3d6r
[c] = x ->20and, after the first use, you would just write
d4,d6 [ ] x d6 – d6 +1,10 [ ]
until the content of the square brackets next changed.
Let’s break that example down for anyone who’s struggling to keep up (should be no-one but you can never tell).
If this roll is to be used in place of a d20, you get condition A, in which your main roll is d4 x d6. The “r” in condition [a] signifies ‘replacement’.
If the roll is to be used in place of 3d6, you get condition [b], and the main roll becomes d6 x d6.
Both a and b have a modifier of +1. But if the results of the multiplication and subtraction of die rolls – that’s the “x -” in condition [c] – that is greater than 20, that modifier goes up to +10.
All clear?
So, for the remainder of this subsection, I’ll be writing d4 x d6 – d4 +4,6 [ ] for the die roll, with the [ ] signifying [d10r, 3d6r] without explicitly stating the condition every time. Okay?
An afterthought – how do you decide where the body ends and the tail begins?
There is a sharp flattening out of the curve at the point of division. You may even enter a secondary peak.
Everything to the right of that dividing line is tail, everything to the left of it is body.
Min, Max, Ave
Minimum = [1, 3]
Maximum = [27, 29]
Average = [10.25, 12.25]
Right away, the new format has been extended to the display and differentiation of results, showing them in a far more compact way than would otherwise be possible.
I got the average the old-fashioned way – multiply each result by it’s % chance and divide the total of all those results by 100.
I did so because I wanted to test a shortcut that I’ve been using without verification like, forever – substituting in the value of an average roll to calculate the average result of a complex expression like we have here. So let’s try it:
2.5 x 3.5 – 2.5 + 4 =
8.75 – 2.5 + 4 =
10.25correct result. It seemed logical and obvious to me that it would work, but I’ve never actually tested it to be sure, until now.
The Thresholds
The 1% Threshold
Everything beats this – but in the cases of [1 & 22-27, 3 & 24-29], only just, at a probability of 1.04%.
The 3% Threshold
Now things get juicier. in addition to the results mentioned above, each roll has 5 results below this threshold, and they are all in the tail: [16-17 & 19-21, +2]
Another extension to the protocol – instead of explicitly listing the second case results, I’ve just indicated what the difference is.
In the case of the second example offered initially, because the core die roll was changing, this wouldn’t work and you would have to use the longer, more explicit format continually.
The +2 simply indicates, add 2 to get the alternative results, so 16-17 becomes 18-19.
The 5% Threshold
Between 3% and 5% things get more varied. We have one result in the main body – [2, +2] – and the entire rest of the tail except for [12, +2] – [10-11 & 13-15 & 18, +2].
The 10% Threshold
There are no results with a higher probability than this threshold, so the 5-10% bracket holds the entire rest of the results: [3-9 & 12, +2].
I don’t usually do this, but I thought it would be worthwhile this time around: a summary of these results in tabular form.
1-3%: [1, +2]
3-5%: [2, +2]
5-10%: [3-9, +2]
3-5% [10-11, +2]
5-10%: [12, +2]
3-5%: [13-15, +2]
1-3%: [16-17, +2]
3-5%: [18, +2]
1-3%: [19-27, +2]
The other thing worth mentioning is that the average has clearly been ‘pulled’ to a higher number by the tail. If [3-9,+2] is considered the main body, which is what the above results show, then you would expect an average of [6, +2] or thereabouts.
The greater the probability contained in the tail, the greater the shift. In this case, up a full 4.25 from [6,+2] to [10.25,+2].
That will have an impact in the next section.
Slices Of Range: Percentages Of Probability
Range Of Results
27-1 = 26, +1 for the 1 itself, makes 27.
The results are [,+2] higher for the alternative construction, but the range is exactly the same.
Ave – Min, Max – Ave
10.25 -1 = 9.25
27 – 10.25 = 16.75
Because minimum, maximum, and average all go up by the same amount in the second formulation, these ranges are exactly the same.
The tail isn’t quite twice as long as the main body – 16.75/9.25 = 1.8108. I’ve never tested whether or not that’s true globally, so at this point it’s just an observation, not even a demonstration of a rule-of-thumb principle..
1/3 (Ave-Min) + Min
The part of the graph that lies below the average is going to take in the entire body and part of the tail.
[1/3 x 9.25 + 1 = 4.0833, +2]
So the band of worst results runs from [1 to 4,+2] and has a combined probability of 17.71%.
2/3 (Ave-Min) + Min
[2/3 x 9.25 + 1 = 7.1667, +2]
The poor results are from [5 to 7,+2] and these have a probability of 42.71 – 17.71 = 25%. So 1 in every 4 rolls will yield a [5, 6, or 7,+2].
The Lower Core
This obviously contains everything else up to the average, so [8-10,+2]. The total probability of these results is 59.38 – 42.71 = 16.67%. This is ever-so-slightly less than the bottom band.
The Upper Core: 1/3 (Max-Ave) + Ave
For the first time, we have an asymmetric roll, which means that I can’t simply echo the spans in reverse sequence, I have to actually calculate these values.
[1/3 x 16.75 + 10.25 = 15.8333,+2]
So the upper core is 11-15, and includes the secondary peak at 12. The total probability in this span of 5 results is 80.21 – 59.38 = 20.83%.
If the main body is 3-9, this shows that the early part of the tail is quite fat.
2/3 (Max-Ave) + Ave
[2/3 x 16.75 + 10.25 = 21.4166,+2]
The band of ‘good’ results ranges from [16 to 21,+2] and has a total probability of 93.75 – 80.21 = 13.54%.
This is the lowest-probability band that we’ve see so far. But the 93.75% [1-21,+2] indicates that there’s not much probability left for the very best results.
The Lofty Outcomes
The results from [22-27,+2] have to contain the rest of the 100% total, so 100 – 93.75 = 6.25%.
d4 x d6 – d4 +4,6 [ ]:
[01-04,+2]: 17.71%, span 4, sub-average=4.4275%
[05-07,+2]: 25%, span 3, sub-average=8.3333%
[08-10,+2]: 16.67%, span 3, sub-average=5.5555%
[11-15,+2]: 20.83%, span 5, sub-average=4.166%
[16-21,+2]: 13.54%, span 6, sub-average=2.5667%
[22-27,+2]: 6.25% span 6, sub-average=1.0417%
This table introduces a new diagnostic tool, the sub-average. This is the probability of the range divided by the span of results – so the range of [05-07,+2] has a total probability of 25% and a span of 3, giving an average probability across the span of 8.3333%.
The combination of range and sub-averages gives a very approximate description in actual numbers of the shape of the probability curve, ironing out little deviations like the secondary peaks at [12 and 15 and 18,+2].
I haven’t needed it before, but this is a far more complicated curve than the previous ones.
Slices Of Probability: The Definitive Result Values
The Lowest 20%
The 20% mark in total probability falls between [4 and 5,+2], so this band runs from [1-4,+2].
Second Lowest 20%
The 40% mark is a little below 7, so this 20% holds results from [5-6,+2].
The Middle 20%
We get a total probability of 60% just above [10,+2], so this band contains results from [7-9,+2].
Second-Highest 20%
The 80% total is reached just below 15, so this group contains results [10-14,+2].
Highest 20%
Which leaves only the cream of the crop, from [15-27,+2].
Peak Probability
The peak probability belongs to a result of [6,+2], exactly as I forecast from the body range of [3-9.+2]. It is 9.38%.
Matching Result: 1/3 Peak Probability
1/3 x 9.38 = 3.1267%
[2,+2] equals this almost exactly, at 3.13%. It’s so close that it has to be included.
In the tail, things get more interesting. You can look at the probability chart and describe the tail as having peaks at 12, 15, and 18, and/or you can talk about valleys at [10-11, 13-14, and 16-17,+2].
[14 & 16-27] are all below this threshold.
Matching Result: 2/3 Peak Probability
2/3 x 9.38 = 6.2533%.
[3 & 8-13 & 15,+2] are all at or below this value.
Which leaves [4-7,+2] as exceeding it.
The question is always whether or not results that land exactly on a dividing line like this should be counted above or below it. But in this case, [2,+2] above set a precedent of including such cases in the lower of the divisions. So the dividing lines can be read as “[value] or less”.
d4 x d6 – d4 +4,6 [ ]
01-20%: [1-4,+2], span 4
21-40%: [5-6,+2], span 2
41-60%: [7-9,+2], span 3
61-80%: [10-14,+2], span 5
81-100% [15-27,+2], span 13
[1-2,+2] 4.17%, span 2
[3,+2] 5.21%, span 1
[4-7,+2] 33.33%, span 5
[8-13,+2] 30.30%, span 6
[14,+2] 3.13%, span 1
[15,+2] 4.17%, span 1
[16-27,+2] 19.79%, span 12
Summary Of Results
This is about as simple and clean as a multiplied die roll gets. The addition or subtraction of a die has done it’s job.
If you examine the d4 x d6 chart above, one of the first things you notice is that it looks unfinished and incomplete. There are gaps – there’s no way to roll a 7, for example. Adding or subtracting a die fills in those gaps – at the expense of lowering probabilities (the possibility of the additional results ha to come from somewhere).
Note that if the gaps are too large, a d4 might not be big enough. With d6 x d8 – d4, there is still a gap between 41 and 44, with two results missing. To fill them in, the d4 has to grow to a d6 – note that 6-4=2=the number of missing results.
When To Use This Substitute
- Unhosted Half-life
- Base Infectious Rate;i>
- Immunity
- Pre-symptom period
- Infectious Stage Start
- Infectious Stage End
- Symptom Recovery
- Disease Recovery
- Unhosted Half-life = 9 days
- Base Infectious Rate = 12/-
- Immunity = 3%
- Pre-symptom period = 4 days
- Infectious Stage Start = 5 days
- Infectious Stage End = 6 days
- Symptom Recovery = 10 days
- Disease Recovery = 7 days
- Determine the chance of failure of 1 exposure.
- Convert it to a decimal.
- Estimate the number of exposures to be rolled at once. 20, 50, 100 – the choice is yours.
- Raise the decimalized risk of NOT catching the disease to the power of the number of exposures.
- The result will be a much smaller number. Convert it to a percentage.
- That’s your chance of not contracting the disease. Subtract from 100 to get the matching chance that you WILL contract the disease.
- For example, let’s take our 12/- and assume it’s on 3d6. That’s 74.07%. But 10 half-lives have passed since then; 2^10 = 1024, so the chance per exposure is now down to 74.07 / 1024 = 0.072334%.
- Which means your chance of NOT catching it is 99.927666%.per exposure.
- to convert it to a decimal, divide by 100. So that’s 0.99927666.
- The GM decides that every 100 exposures sounds about right, with each step (and the dust raised) counting as an exposure, as does handling an object, touching a surface, or engaging in a round of combat.
- 0.99927666 ^ 100 = 0.930196.
- 0.930196 = 93.0196%.
- So, every 100 exposures, there is a 6.9804% chance of catching the disease.
- Instead of counting, the GM assumes 100 feet of walking is 100 steps, and whenever the time since the last check feels about right, based on their activities since, he has the characters roll.
- 0.99927666 ^ 500 = 0.696.
- 0.696 = 69.6%. So there’s a 69.6% chance of NOT getting it every 500 exposures.
- Which means that there’s a 30.4% chance of catching it, per roll.
- Unhosted Half-life = 2 days
- Base Infectious Rate = 18/-
- Immunity = 6%
- Pre-symptom period = 15 days
- Infectious Stage Start = 15 days
- Infectious Stage End = 18 days
- Symptom Recovery = 8 days
- Disease Recovery = 14 days
- 18/- on d20 = 90%
- 14 half-lives so 90 / 351844 = 2.55795e-4%
- 100 – 2.55795e-4 = 99.999744205%
- 99.999744205% = 0.9999744205
- 500 checks
- 0.999744205 ^ 500 = 0.9987218
- 0.9987218 = 99.87218%
- 100 – 99.87218 = 0.128159%. Effectively no chance.
- 5500 checks – the entire dungeon: 0.999744205 ^ 5500 = 0.986
- 0.986 = 98.6%
- 100 – 98.6 = 1.4%
I wouldn’t use this to replace a d20 or 3d6 rolled for the usual purpose. I WOULD use it to replace those things on a custom table.
For example, when it comes to diseases, there are all sorts of things that you need to know.
Now, you could get some graph paper and draw a number of pretty curves to represent the probability you want; total those up and you can scale to exactly 100%.
Or you can simply use a die roll like this one to create the curves for you.
If I were to do that, I might get:
All these numbers were generated just by rolling d4 x d6 -d4 +4.
What do these numbers mean? Well, a disease starts out with a Base Infection Rate chance of being caught. If it’s out in the open, in the soil for example, it loses half it’s infectiousness as disease cells die off every unhosted half-life that passes.
So it starts as 12/- – that could be on 3d6 or d20 or whatever. After 9 days, it’s down to 6/-. 9 days later, it’s 3/-. 27 days later, and it’s 1.5 /-; then 0.75, 0.375, and so on. But that’s per exposure – if a dungeon was once plagued by the illness, you might easily have 10, or 50, or 100 exposures.
You aren’t going to roll all of them. There’s a shortcut.
If the dungeon is 100′ x 100′, divide the area by 2 – that’s a safe estimate for the minimum number of exposures through the whole thing, without allowing for rounds of combat, touching things, etc. So 100^2 / 2 = 10000 / 2 = 5000 exposures. Every 100 exposures means 50 checks will be needed. The GM decides that’s too many and decides to increase the number of exposures per check to 500.
A 30.4% chance per check, 10, maybe 11 checks, 4 PCs – what are the odds?
At least 1 character: 100%. Well, more than 99.9999%.
At least 2 characters: 99.997%.
At least 3 characters: 99.80%
All four characters: 92.78%
Once someone is infected, they no longer need to roll, but the GM doesn’t want them to know that anything’s changed until symptoms appear, so he lets them continue and just ignores the results.
The pre-symptom period is 4 days, so 4 days after infection, the symptoms start.
A day later (5 days), the character becomes infectious – at the full base rate of 12/- on 3d6.
They will stop being infectious 6 days later, so 11 days after infection (5+6=11).
Symptoms might end before or after that date. The disease is far more dangerous if they end while the sufferer is still infectious! But in this case, symptoms persist for 10 days, so they end on day 14 (4+10=14). For the last 3 days of that period, they were no longer contagious.
But the disease will have taken it’s toll. Recovery was rolled at 7 days, and that final clock starts when all the others have stopped – so day 21 is when the victim is back to their old selves – assuming they survived.
In all 8 cases, the roll used was this one, and the results then interpreted. If I rolled up a second disease the same way, the results would be completely different:
This is a much slower, more pernicious ailment – but despite it’s very high infectiousness (18/- on d20 this time), it has a very short half-life, and 45 of them have passed..
So, unless there are 50 people in the party, it’s extremely unlikely that anyone will catch this. It’s half-life is so short that’s effectively dead. But encountering someone who has managed to beat those odds would be extremely bad news. 18/- on d20 chance of catching it? And not knowing it until 15 days later?
Depending on your interpretation of the rules, having a disease like this might mean that ‘Cure’ spells no longer work on you – that they try to cure the disease and fail. If that’s your GM’s interpretation, it might at least offer an early clue.
On the other hand, at least part of hit points are self-confidence, and there would be a psychological lift at receiving a Cure Light Wounds spell, and cosmetic improvements, so you might well regain some HP, anyway.
Okay, here’s the important bit: Why this die roll works
As the analysis shows, results skew markedly low. It’s rare for anything to be higher than 12. But it can happen. That means that results are focused on the trait that you are rolling for, and need only simple interpretation.
While it’s rare to get a high result, it can occasionally happen, and it always causes something memorable and significant when it does.
A couple of quick other notes about multiplied die rolls.
- If you want the curve to bias in the other direction, (Maximum+1) – the die roll is your solution.
- There’s a huge temptation to try dividing a something by the die roll. Don’t – it’s impossible to control. Most of your results will be sensible, but there’s always going to be a divide by 1 or a divide by zero to mess things up.
Exotic Choice #9: d30 +1 – d10
This roll looks deceptively simple. It only has two dice, for heaven’s sake!
And yet, dice subtraction can sometimes do weird things, so let’s take a look at this one…
Probably the first thing you notice is the flat top of what might once have been a triangle. It runs from 1 to 21.
The second thing that strikes you is the enormous range of results – from -8 to 30.
And then that minimum result sinks in. What does a roll of -8 even mean?
Min, Max, Ave
Minimum -8.
Maximum 30.
Average 11.
The Thresholds
The 1% Threshold
-6 is exactly at the 1% threshold. So is 28. So the really improbable rolls are -8 to -6 and 28-30.
The 3% Threshold
0 and 22 are exactly at the 3% threshold – so the unlikely rolls are -5 to 0 and 22 to 27.
The 5% Threshold
Nothing gets this high. Everything from 1 to 21 is at an absolutely flat 3.33%.
Slices Of Range: Percentages Of Probability
Range Of Results
30-(-8) = 38, +1 for the -8 result itself. So there is a span of 39 results!
Ave – Min, Max – Ave
11-(-8)= 19.
30-11=19.
So the roll is symmetric. The fact that the range spans an odd number of results means that there will be one result nominally in the middle whose probability is going to have to be split.
1/3 (Ave-Min) + Min
1/3 x 19 + -8 = -1.6667, so the division falls between -1 and -2.
So the lowest division of results runs from -8 to -2, and comprises 9.33%. Span of 7.
2/3 (Ave-Min) + Min
2/3 x 19 + -8 = 4.6667, so the division is between 4 and 5, which means that the next tier of results are -1 to 4. These have a total probability of 28.33 – 9.33 = 19.00%. Span of 6.
The Lower Core
-
That means that everything from 5 to 10, and half of 11, form the lower core. This group have a total probability of 48.33 – 28.33 + 1/2 x 3.3333 = 20 + 1.6667 = 21.6667%. Span of 6 1/2.
The Upper Core: 1/3 (Max-Ave) + Ave
The upper side is a mirror-image of the lower. So the upper core is 6 1/2 wide, including 11 (which is split). That gives results of 11-17 and total probability of 21.6667%.
2/3 (Max-Ave) + Ave
Above the central core are the good results, a span of 6, starting at 18 – so 18-23 – and with a probability of 19.00%.
The Lofty Outcomes
At the very top, the very best results therefore are 24-30, a span 7, and a total probability of 9.33%.
d30+1 -d10:
-8 to -2 = 9.33%, span 7.
-1 to 4 = 19%, span 6
5 to 11 = 21.667%, span 6.5
11 to 17 = 21.667%, span 6.5
18 to 23 = 19%, span of 6
24 to 30 = 9.33%, span of 7.
Slices Of Probability: The Definitive Result Values
The Lowest 20%
The 20% total comes between 1 and 2, so -8 to 1.
Second Lowest 20%
The 40% mark is reached between 7 and 8, so this bracket contains 2-7.
The Middle 20%
We cross the 60% mark between 13 and 14, so this band consists of results from 8-13.
Second-Highest 20%
80% is almost but not quite to the 20 result. So this band contains 14-19.
Highest 20%
Which obviously leaves results from 20-30 to form the highest band of results.
Peak Probability
As already mentioned, this is 3.3333% – and it’s shared by 21 results.
Matching Result: 1/3 Peak Probability
1/3 of 3.3333 = 1.1111%. Results of -8 to -6, and 28 to 30, are below this level.
Matching Result: 2/3 Peak Probability
2/3 of 3.3333 = 2.2222%. That probability band contains -5 to -3 and 25 to 27.
Everything else, from -2 to 24, is between 2.2222% and 3.3333%.
d30+1-d10:
01-20%: -8 to 1, span 8.
21-40%: 2 to 7, span 6
41-60%: 8-13, span 6
61-80%: 14-19, span 6
81-100% 20-30, span 11
-8 to -6, 2%, span 3
-5 to -3, 5%, span 3
-2 to 24, 86%, span 27
25 to 27, 5%, span 3
28 to 30, 2%, span 3
Summary Of Results
If you use the full span of results, you are going to get some very extreme results. But here’s the thing: If you re-roll any result below 1 or higher than 20, this is a perfect d20 simulation.
Of course, it’s a lot of malarkey to go through for that result.
When To Use This Substitute
This is the perfect die roll for bringing a sense of the absurd or ridiculous into a game. For example, when two combatants are roaring drunk.
Anytime someone rolls below 1, they do something stupid or something completely ridiculous happens to them. Anytime someone rolls above 20, something ridiculous happens to their opponent.
When circumstances warrant neither a farce nor a circus, there are better constructions to choose. But when those are the orders of the day, this construction is hard to beat.
Exotic Choice #10: 5d4 / d5
We’ve had multiplication and subtraction as well as the more commonplace addition – so it’s no surprise that division makes an appearance at this point.

This chart shows three curves,
all discussed in the text below (or this caption would be far too long):
5d4 / d5; [(2d4+2d6+2d8) / 3d2] +1; and (6d4 / d6) +5.
Three compositions for the price of one!
The first, 5d4 / d5, is the one we’re mainly interested in. It shows all of the classic characteristics of a divided die roll quite clearly – there’s a front, a crown, a back, and a tail with a secondary peak or ‘hump’.
The second shows how complicated these things can get. It was chosen to illustrate two things, maybe three: (1) that 2d4+2d6+2d8 have a maximum of 36, the same as 6d6; (2) that if the denominator is large enough with respect to the numerator, the ‘crown’ can compress into a single point with an extremely high probability – note the scale on the left and you’ll see that the peak is approaching 30% probability. That’s absolutely ridiculous in a roll with this many results! And (3) the back can make a smooth descent to a long tail of virtually no probability while the ‘hump’ has been flattened out of existence, so this shows how the shape of a divided die-roll curve can change.
The third is the configuration I almost chose for this section, shifted 5 spaces to the right because the resulting curve is so like the subject one that it would be confusing. But now that you can see how similar they are by having them side-by-side, you can meaningfully evaluate the differences, which are also significant in revealing traits of divided-die-roll anatomy:
First, notice that the brown line – our subject construction – isn’t quite flat at the crown, and that our reference comparison, the gold line, is even more angled. I’ve never seen one slope the other way, but that wouldn’t surprise me if I did.
Second, notice that the gold reference line has a tertiary hump at results of 8 & 9 – and, in fact, that our subject composition has one too, at 6 – it’s just a lot smaller.
Until I saw just how similar they are, I was tossing up whether or not to include 6d4 / d6 as a bonus extra, even though time is growing a little short and there’s still a lot to do. But the differences seem to be so small that it’s not worth the effort, and time, involved.
Afterthought: How do you decide where the back ends and the tail starts?
As with the multiplied die roll, there is a sudden flattening, and maybe even entry into a secondary peak. The back includes any tertiary hump(s).
In this case, 4 is a transition between crown and back; 5 is back; 6 is back and the tertiary hump; 7 is back; 8 is back; but at 9, there is a flattening, and 10 starts the buildup to the secondary hump in the tail. so 4-8 are the clearly back and 10+ are clearly tail, with 9 the dividing point, able to go either way.
From the definitions, and comparing the probability differences 8-to-9 (0.98%) to that from 9-to-10 (0.33%), there is an obvious difference that connects 9 more strongly to the tail than to the back. So I would classify 9 as the start of the tail.
Min, Max, Ave
Minimum 1
Maximum 20
Average predicted 5 x 2.5 / 3 = 4.1667
Average, measured = 5.4367 (which makes me glad that I decided to do it both ways!)
The Thresholds
The 1% Threshold
The only results with a probability of 1% or less are in the end of the tail, from 17-20.
The 3% Threshold
This threshold is a bit more diverse. Falling beneath it are the front (1), a little of the back (8) and most of the tail (9-11, 14-16).
The 5% Threshold
Between 3% and 5% there is part of the back (7) and the rest of the secondary hump in the tail (12-13). In fact, half the time, I would probably have rounded the latter (3.03%) down to include them in the 1-3% category. But the more accurate approach better reflects the anatomy of the die roll results.
The 10% Threshold
The 5%+ to 10% bracket has the middle of the back (5-6).
The 15% Threshold
In this bracket we have the remainder of the back (4).
The 20% Threshold
Both results in the crown climb higher than this percentage (2-3) – which will result more than 40% of the time, collectively!
5d4 / d5:
1% to 3%: 1
20 to 25%: 2-3
10% to 15%: 4
5% to 10%: 5-6
3% to 5%: 7
1% to 3%: 8
1% to 3%: 9-11
3% to 5%: 12-13
1% to 3%: 14-16
1% /-: 17-20
Slices Of Range: Percentages Of Probability
Range Of Results
There are 20 results, so if the curve were symmetric (it’s not) there would be two results with equal probabilities in the crown.
Ave – Min, Max – Ave
Here’s where things get interesting!
5.4367 – 1 = 4.4367.
20 – 5.4367 = 14.5633.
One side of the average result is more than 3.2 times the size of the other!
1/3 (Ave-Min) + Min
The worst results band runs from the minimum (1) to
1/3 x 4.4367 + 1 = 2.4789 – so almost exactly mid-way between 2 and 3.
1-2 have a total probability of 23.75%.
2/3 (Ave-Min) + Min
2/3 x 4.4367 + 1 = 3.9578, so 4 doesn’t quite> make the cut – but it’s so close that I would round to include it, anyway, splitting it in two (a leg in both camps).
3 has a probability of 21.25%, +1/2 of 4’s probability of 13.01 = 6.505%, gives a total of 27.755%.
The Lower Core
Between 3.9578 and 5, we have 5, and the other half of 4. 5 has a probability of 8.57%, and 1/2 of 4 is still 6.505, so the total probability here is 15.075%.
The lower bands of the curve total 66.58% of all the results!
The Upper Core: 1/3 (Max-Ave) + Ave
1/3 x 14.5633 + 5.4367 = 10.2911333, so the upper core stretches from 6 to 10 – that’s the lower back and the start of the tail, but not including the peak of the secondary hump.
6-10 have a total probability of 84.34 – 66.58 = 17.76%.
2/3 (Max-Ave) + Ave
2/3 x 14.5633 + 5.4367 = 15.1455666, so 11 to 15 make up the ‘good but not great’ band of results. Those have a combined probability of 97.64 – 84.34 = 13.3%.
The Lofty Outcomes
That leaves the great results as being 16-20, with a combined probability of 100 – 97.64 = 2.36%.
But I want to especially note the low chance of a 20 at 0.02%. Rounding error is likely to be huge, but on the face of it, you are 5 / 0.02 = 250 times more likely to get a 20 on a d20 than on this roll.
A moment’s reflection will show why – to get there, absolutely everything has to go right. Maximum result on the 5d4 (20) and minimum result on the d5 (1). Out of 5 x 4 x 5 = 100 possible results. Actually, by my math, that’s a 1% chance, so I’m going to have to look into this a little further. One moment…
(a few minutes later:) Okay, I’m back. My mistake in the above is in calculating the number of possible outcomes on the 5d4, which I’m sure most of you will have spotted right away.
The correct number of possible result combinations of die faces is 4^5, not 4×5. That gives 1024, which multiplied by 5, gives 5120 combinations all told. Only 1 of them produces a result of 20, so that’s 0.01953%. And a d20 does indeed have 256.016385 times greater likelihood of resulting in a 20.
All this might seem like a minor side-note at the moment, but I’m thinking ahead, and expecting it to weigh heavily on evaluating when to use this particular construction.
5d4 / d5:
1-2 ‘Worst possible roll’ = 23.75%, span 2
3-4 ‘Poor result’ = 27.755%, span 2
4-5 ‘Below Average result’ = 15.075%, span 2
6-10 ‘ Above Average result’ = 17.76%, span 5
11-15 ‘Good result’ = 13.30%, span 5
16-20 ‘Great result’ = 2.36%, span 5
(20 ‘Best possible result’ = 0.01953%).
Slices Of Probability: The Definitive Result Values
The Lowest 20%
We get to the 20% total really quickly – in fact, only one result falls into this band, a 1, which has a probability of just 2.79%. Extending the range to 2 carries it over the 20% total, to 23.75%.
That tells me two things: (1) this tool is of limited utility for the analysis of divided die rolls because of the phenomenally steep face and high crowns; and (2) it might still be useful if I round and generalize a bit. This will compromise the precision of the result, but still give some value in terms of understanding the die roll.
So, on that basis, the ‘lowest 20%’ contains 1-2.
Second Lowest 20%
And, right away, that plan goes off the rails and for exactly the same reason. The 40% mark lands between 2 and 3 and 2 has already been used – so that depopulates this entire zone. I could round 3’s 45% total down to include it, I suppose, but 45% is a full quarter of the way through to the next band.
Part of the purpose in breaking up all these rolls in the same size divisions – the 20%’s – was to enable direct comparison. (dividing the range of results into two parts about the average and each part into thirds has a similar comparative benefit but one arranged around the results, not the probabilities). That still has value, so I’m going to accept the rounding.
Which means that this band consists of the result of 3.
The Middle 20%
The 60% mark is between 4 and 5, so this band also contains just one result: 4.
Second-Highest 20%
We get to 80% almost exactly at 8 – we’ve had to swallow much larger deviations twice already than including 80.68 in the 61-80% band – so this is 5-8.
Highest 20%
Which leaves 9-20 for the rest. Basically, anything in the tail is a ‘good result’ to some degree.
Peak Probability
This belongs to the result of 3, at 21.25%, which narrowly beats 2 and 20.96%.
Matching Result: 1/3 Peak Probability
1/3 x 21.25 = 7.0833. That point-0833 can be very important because it makes it almost impossible for any result to fall exactly on the line, which is more likely to happen with an exact integer result.
Anyway, 1 and 6-20 all fall below this line, with no results close enough to 7% to even argue about.
Matching Result: 2/3 Peak Probability
2/3 x 21.25 = 14.1667. Again, a clear division between the results – 4-5 are below this line and 2-3 are above it.
5d4 / d5:
01-20%: 1 to 2, span 2.
21-40%: 3, span 1
41-60%: 4, span 1
61-80%: 5-8, span 4
81-100% 9-20, span 12
1: 0-7%, span 1
2-3: 14%+, span 2
4-5: 7-14%, span 2
6-20: 0-7%, span 15
Summary Of Results
This is a fairly basic divided die roll. It exhibits all the traits of that type of construction. It’s massively biased low in results, with a long tail of relatively low probability. You can spend hours playing around with variations of the general principle, and often land on unexpected results.
When To Use This Substitute
This is the die substitute to use when failure is – in the GM’s mind – not possible, but degrees of success and complications of pathway in getting to that success ARE.
“So, you’ve rolled a 2? No problem, here’s what happens…” followed by set-back after set-back, and a last-minute success that the characters fall into more than reach towards. In other words, it’s all about driving the narrative, about roleplay.
And if you should happen to fall over the line with a result in the tail, that indicates one of those occasions where the universe seems bound and determined to let you succeed; even outright errors of judgment end up working to your benefit, potentially earning the party an unjustified reputation for brilliance – which they will then have to try to live up to.
Exotic Choice #11: (3d6+2) / d4
Having examined the probability curve, this construction has only one novel feature – a singular peak of probability at result 3. So I’ve decided that it’s not worth the additional time it would take, which I can put to better use on something far more exotic and interesting.
Exotic Choice #12: (4d10 / 2) – d2 +1
Okay, now this one’s subtle. If you look really closely, i think there’s the most minute difference in the two sides of the curve. To test this perception, below are graphed two curves: the Main Curve, M, and 21-M.
If there is a difference, the two will not line up.
The main roll is averaging just a little higher probability on the low side of the average and a little less on the high side. I wonder what that will do to the average?
Min, Max, Ave
Minimum 1
Maximum 20
Average: Predicted: (4 x 5.5) / 2 -1.5 + 1 = 11 – 1.5 + 1 = 10.5
Average, measured:10.24977502
Call it 10.25. And there, again, is that very small difference manifesting itself.
So I decided to look into why it’s there. Here’s what I found: The division by 2 implicitly rounds down results by treating odd and even rolls on the 4d10 differently. What appears to be one curve is, in fact, the sum of two interleaved curves – odds and evens. Because we’re dividing by 2, the losses on the odd-result rolls are -0.5 each, and because half the possible results of 4d10 are odd and half are even, when this gets averaged over the whole, the net effect is a -0.25 bias on the results. It’s a perfect example of how small nuances can manifest in real-world differences.
The Thresholds
The 1% Threshold
Below this threshold are 1-3 and 17-20, so the overall shift low has already had a significant effect. 1-3 have a cumulative probability of 0.80% (so they are well below the 1% mark individually), while 17-20 have a 1.4% total probability – but span 4 results, not 3.
The 3% Threshold
4-5 and 16 are below the 3% threshold. The difference in span is because the curve almost has symmetry, meaning that any disparity is likely to be counterbalanced somewhat later on. In this case, the previous band’s spans of 3 vs 4 are the disparity, and the difference in spans, 2 vs 1, this time around are the counterbalance.
4-5 have a total probability of 4.42%.
16 has a probability of 2.13%.
The 5% Threshold
6 is almost at the 5% threshold, with a probability of 5.08%. It’s close enough for my money. At the high end, we have 15 & 16 – so the disparity in spans returns. 15-16 is a combined probability of 6.07%.
The 10% Threshold
7 on the low side and 13-14 on the high are in the 5-10% bracket – so the disparity has worsened.
7 has a probability of 7.63%, while 13 & 14 total 15.24%.
The 15% Threshold
Everything that remains is in the 10-15% range, nothing breaks the 15% threshold. So that’s 8-12, which have a total probability of 38.02%.
With no time left to even out the disparity, it has to stand – meaning that the right-hand side is cumulatively down on probability and needs a longer span to get to similar probability values. The span of this central region is 5 results.
(4d10 /2) -d2 +1:
<1%: 1-3 = 0.8%, span 3
1% to 3%: 4-5 = 4.42%, span 2
3% to 5%: 6 = 5.08%, span 1
5% to 10%: 7 = 7.63%, span 1
10% to 15%: 8-12 = 38.02%, span 5
5% to 10%: 15.24%, span 2
3% to 5%: 15-16 = 6.07%, span 2
1% to 3%: 16 = 2.13%, span 1
<1%: 17-20 = 1.4%, span 4
Slices Of Range: Percentages Of Probability
Range Of Results
Results span from 1 to 20, so a range of 20.
Ave – Min, Max – Ave
10.25 – 1 = 9.25.
20 – 10.25 = 9.75
There, once again, is the very subtle asymmetry lurking in the heart of this construction. At least I know and understand what’s causing it now.
1/3 (Ave-Min) + Min
1/3 x 9.25 + 1 = 4.0833, so 4 just scrapes into the lowest division of results. 1-4 have a total probability of 2.28%, roughly half of which is 4, and half of what’s left is 3. The remaining quarter is split between 1 and 2.
2/3 (Ave-Min) + Min
2/3 x 9.25 + 1 = 7.1667, so this band contains results from 5 to 7. They have a collective probability of 17.93 – 2.28 = 15.65%. That’s 6.864 times the probability of the previous division, meaning that you would expect to see 5, 6, or 7 come up about 7 times for every result in the 1-4 range.
The Lower Core
8-10 fall into this band. They have a combined probability of 53.30 – 17.93 = 35.37%.
That’s about 2 1/4 times the probability of a 5-7 result, so for every four results in that range, you would expect to see 9 rolls producing results of 8-10.
The Upper Core: 1/3 (Max-Ave) + Ave
Because of the asymmetry, this has to be actually calculated.
1/3 x 9.75 + 10..25 = 13.5. This range contains results from 11-13, and they have a combined probability of 86.22 – 53.30 = 32.92%, just a little less than the lower core.
In fact, while probability says that it could happen sooner, what this amounts to is 12 results in this span for every 13 in the lower core.
2/3 (Max-Ave) + Ave
2/3 x 9.75 + 10.25 = 16.75. This band contains results from 14-16, which have a combined probability of 98.6 – 86.22 = 12.38%.
The upper core will result 2.7 times as often as this range, so for every 8 results in the above average category, there will be 3 ‘good’ rolls.
The Lofty Outcomes
The best range of results are therefore 17-20, with a combined probability of just 1.4%.
That’s a ratio of 8.8 times, so for every 5 results yielding this tier, there would be 44 rolls of the band below it.
Slices Of Probability: The Definitive Result Values
The Lowest 20%
20% of the probability contains results from 1-7 – so, on 100 rolls you would expect to see 20 of them within this range, give or take.
Second Lowest 20%
The 40% mark just fails to capture 9, so results of 8, technically, have this 20% all to themselves. That said, 40.19% is close enough that I’ll include it here for a span of 2. I think that’s a fairer representation of both results.
The Middle 20%
The 60% mark splits the difference between 10 and 11, magnifying the asymmetry to the point where it is undeniable – 60% of the rolls will be below the average, and 40% above it.
10 alone occupies this space, with an actual probability of 13.12%. When the disparity is that large (13.12% vs 20%, so almost half of the 20% is missing), you have to consider including the next result up. 11/- has a combined probability of 66.08%, so this would be an error – but it’s a smaller error than not doing it. So this 20% is now considered to be 10-11, and to have a span of 2.
Second-Highest 20%
The 80% mark is distinctly between 12 and 13, so this range contains a single result 12. However, 12 only has a probability of 11.22% – even closer to 1/2 of the desired range of results. So I have to look at whether or not 13 can be included, with it’s 8.91% probability. The combination is a total of 20.13%, so even without looking at the combined value, I’m inclined to say yes. That combined value of 86.22, as before, does represent an error, but it’s a smaller error than not doing so, which confirms the predisposition. So this band is 12-13, a span of 2.
Highest 20%
But that leaves the last 20% to hold everything else – results from 14 to 20. That’s a span of 7 results, which is the same size as the first bracket, to be fair.
Peak Probability
Breaking this down by the alternative route requires the Peak probability. This belongs to a result of 10, without question, and 10 has a probability of 13.12%.
Matching Result: 1/3 Peak Probability
1/3 x 13.12 = 4.3733%.
1-5 are below this chance, and so are 15-20. note: span of 5 and span of 6, respectively.
Matching Result: 2/3 Peak Probability
2/3 x 13.12 = 8.7467%.
6-7 and 14 are below this result. I’d like to have included 13 or 15 to preserve the symmetry in this range, but the error that results is too great. Which means that this range cancels out the span discrepancy of the previous set of results.
That leaves 8-13 as having the highest individual probabilities.
(4d10 / 2) – d2 + 1:
01-20%: 1 – 7, span 7
21-40%: 8 – 9, span 2*
41-60%: 10-11, span 2*
61-80%: 12-13, span 2*
81-100% 14-20, span 7
The results of all that hand-tweaking of errors (indicated by the * in the table above) is a perfect reflection of the underlying symmetry of the curve; the bias is completely hidden. That’s why I’ve used so many analysis approaches – you can never tell which ones will definitively describe the curve, and they are all valid – just with a different emphasis.
1-5: 0-4.37%, span 5
6-7: 4.37-8.75%, span 2
8-13: 9.75%+, span 6
14: 4.37-8.75%, span 1
15-20: 0-4.37%, span 6
More than any other tool, this shows that this curve contains 3 major bands of results – low, middle, and high – connected by two short and therefore steep rises and falls in probability. It’s a classic bell curve, in other words. But it also highlights that slight bias low.
Summary Of Results
And that sums up this construction, really – a classic bell curve with a hidden tiny bias.
When To Use This Substitute
- For the difference between this construction and 3d6 to matter, you need to be making a lot of rolls, or the bias won’t show up.
- The range runs from 0 to 20, but the most extreme values are so unlikely that the practical range is 4-17. So you have to want to have the chance at a more extreme result, whilst making that chance vanishingly small.
- It’s probably fair to say that this is a more ready substitute for 3d6 – but that’s not a good thing as it means you need a compelling reason to make that substitution.
- Substituting this for a d20 roll integrates all the consequences of a bell-shaped curve, so that’s a more dramatic and potentially useful difference – but there are better choices for those cases. You need some valid reason for those choices not to work and for this choice to still be valid in order to justify using this roll. And that’s going to be rare.
To be honest, I can’t think of an occasion that ticks all the boxes for using this alternative. Let’s check off the criteria, though, in case you are cleverer in this respect than I.
Ultimately, I think the greatest value that this construction holds is as an object lesson and a demonstration of principle.
The object lesson relates to subtlety and nuance, and the dangers of making assumptions when probabilities are involved. They can, and from time to time will, lead you astray.
And the demonstration of principle relates to what happens when dividing a die roll by a fixed value. The more you dig into this, the more you get swamped by minutia becoming relevant characteristics.
Dividing by 3, for example, means that 2/3 of results will have a rounding distortion.
Dividing by 4 takes that up to 3/4.
But – some of those bias errors will be larger than others. Take dividing by 10 – a rounding bias that loses 0.1 is not very large, while a bias that loses 0.9 is comparatively huge. And the overall impact: A bias adjustment of -0.5.
Compare that with the divide-by-3: some results will have an error of -1/3, some of -2/3, and some will have no error at all. And the average of -1/3 and -2/3? It’s -0.5 – again.
Dice with unequal numbers of odd results vs even can amplify or diminish the bias slightly. That requires the d# to be odd – so d3, or d5. d7 if you can find them – the only one’s I’ve seen are marked with the days of the week. And so on.
Is the resulting bias large enough to justify the complexity of the process and analysis? I’m not sure that it is, but can’t say that it isn’t either.
BONUS EXTRA: Exotic Choice #13: [log (base 2) [(d6 / 3) ^ d8] +d8 +1}
I’ve saved the weirdest till last! The ultimate in weirdness, this possibility came to me at the last possible moment, just a day before posting this work..
I couldn’t fully analyze this on my own (lack of time more than anything else), so I sought help from Google’s Gemini.
And AnyDice doesn’t understand logarithms, though it does understand exponents. So I’m going to have to do all the analysis the hard way, using a spreadsheet. Which will take additional time.
It’s even possible that I won’t have time to write it up before publication – in which case, I’ll update the post on Thursday and you can read all about this weirdie on Friday.
In which case, right now, the article will shift into ‘conclusions’ mode – but when people check back, they will find this final section miraculously inflated with content!

Have you ever seen anything like it? It looks like some sort of geological formation – but it only uses 3 dice!
The power of this lies in the d6. How well you roll on it determines what effect the first d8 has on the total. If it is 1-2, then the d8 takes a small value and potentially makes it much smaller; if it’s 3 or 4, the effect is neutral; and if the result is 5-6, it takes a large value and makes it much larger – depending on what you roll.
The logarithm then compresses the results back down to a usable scale while placing emphasis on low results.
The second d8 smooths the curve a little, and fills in any gaps, while the +5 shifts the curve into the result space we want.
But it’s by far the weirdest computed probability curve that I’ve ever seen.
NOTE that you can take results of off the table and replace them with results of 20 just by increasing the modifier from +5 to +6.
Min, Max, Ave
Minimum 1
Maximum 21
Average (measured) 6.67989
The Thresholds
The 1% Threshold
Two results fall below this line: 19and 21.
The 3% Threshold
2-3 and 17-18 and 20 are all below this line. 16 is so close to it that I will include it too, at 3.033%.
The 5% Threshold
4-5 and 15 are in this band.
The 10% Threshold
Between 5% and 10% we find everything else – nothing crosses this boundary. So that’s 6-14.
log2 (d6/3) ^ d8 +d8 +5:
1% to 3%: 2-3 = 4.841%
3% to 5%: 4-5 = 6.794%
5% to 10%: 6-14 = 71.942%
3% to 5%: 15 = 4.688%
1% to 3%: 16-18 = 6.894%
<1%: 19 = 0.827%
1% to 3%: 20 = 1.869%
<1%: 21 = 0.276%
Slices Of Range: Percentages Of Probability
Range Of Results
21 results are possible. But with the average so far removed from the mid-point of this range, the roll is biased somewhat low, and that will be reflected in the divisions.
Ave – Min, Max – Ave
6.67989 – 1 = 5.67989
21 – 6.67989 = 14.32011
1/3 (Ave-Min) + Min
1/3 x 5.67989 + 1 = 2.89329. The lower band contains 1 and 2. coming close to inclusion is 3, but not quite close enough.
1-2 have a total probability of 4.03%.
2/3 (Ave-Min) + Min
2/3 x 5.67989 + 1 = 3.78659.
The only result in this span is 3, which has a probability of 2.681%.
The Lower Core
Between 3.78659 and the average are 4, 5, and 6. They have a total probability of 13.106%.
The Upper Core: 1/3 (Max-Ave) + Ave
As usual with an asymmetric roll, this has to be calculated; it won’t be the same as the span on the other side of the average.
1/3 x 14.32011 + 6.67989 = 11.45326, so this band of results contains everything from 7-11, a combined probability of 42.742%.
2/3 (Max-Ave) + Ave
2/3 x 14.32011 + 6.67989 = 16.22663, so this band contains results from 12 to 16, a combined probability of 30.609%.
The Lofty Outcomes
At the very top, we have results 17-21, which have a cumulative probability of 6.833%.
log2 (d6/3) ^ d8 +d8 +5:
1-2 ‘Worst possible roll’ = 4.03%, span 2
3 ‘Poor result’ = 2.681%, span 1
4-6 ‘Below Average result’ = 13.106%, span 3
7-11 ‘ Above Average result’ = 42.742%, span 5
12-16 ‘Good result’ = 30.609%, span 5
17-21 ‘Great result’ = 6.833%, span 5
Slices Of Probability: The Definitive Result Values
The Lowest 20%
1-6 will be the lowest 20% of results.
Second Lowest 20%
The 40% mark captures results 7-8.
The Middle 20%
The 60% mark contains 9-10. 11/-, with a combined percentage of 62.56%, just misses out.
Second-Highest 20%
So it starts this band off, which ends at the 80% mark and a result of 13, capturing 12 along the way.
Highest 20%
Leaving 14-20 as the top end of town.
Peak Probability
The other way of dividing results up is to stratify them by fractions of peak probability, which in this case is 11 at 9.100%.
Matching Result: 1/3 Peak Probability
1/3 of 9.1% is 3.0333%. Below that line we find 1-3 and 17-21, with 16 exactly on the line.
Matching Result: 2/3 Peak Probability
2/9 of 9.1 is 6.0667%. Between this line and the previous one we have a middle stratum of results: 4-5 and 14-15.
Which in turn means that 6-13 are in the uppermost stratum.
log2 (d6/3) ^ d8 +d8 +5:
01-20%: 1 – 6, span 6
21-40%: 7 – 8, span 2
41-60%: 9-10, span 2
61-80%: 11-13, span 3
81-100% 14-20, span 7
1-3: 0-3.0333%, span 5
4-5: 3.0333-6.0667%, span 2
6-13: >6.0667%, span 8
14-15: 3.0333-6.0667%, span 3
16-21: 0-3.0333%, span 6
When you cut the results up this way, the result seems relative prosaic, barely hinting at the complexity below the surface.
You can get even stranger results if you use 3d6/12 as the core roll. Some of the results I got while playing around with the concept looked like a cartoon shark’s tooth!
Summary Of Results
In this case, nothing captures the nuance of what’s going on quite as well as the graph that I made at the top. It’s a bell curve with a flattened top and a longer descent to a secondary peak at 20 – but it’s lumpy.
That’s because this isn’t one curve, it’s the sum of six different curves.
There’s log2 (1/3 ^ d8) +d8,
log2 (2/3 ^ d8) + d8,
log2 (1 ^ d8) + d8 = d8
log2 (4/3 ^ d8) + d8,
log2 (5/3 ^ d8) +d8, and
log2 (2 ^ d8) + d8 = 2d8.
Plus the modifier to shift the results, of course.
When To Use This Substitute
This is the perfect roll to use when results could go either way and snowball, because that’s exactly what is being simulated. The d6 controls the ‘either way’ and the exponentiated d8 controls the degree of snowballing, from none (d8=1) to massive (d8=8). The rest of the construction is just there to make things pretty, and functional.
There is a slight bias low, which is why the average is so low – but that is compensated for because a d6 has an even number of faces, so division by 3 adds a bias high. The result is the tail, which is clearly longer than the front-face of the curve.
The Wrap-up
If the content below looks familiar, it’s because it is, in essence, a summary of the ‘when to use this roll’ discussion, re-sequenced into a more streamlined narrative, with less focus on the die rolls and more focus on the circumstances that suggest their use be considered.
Replacing a d20:
- When you need one and don’t have one to hand, 10 x (d2-1) + d10 or 5 x (d4-1) + d5 are perfect replacements.
- For everyday skill checks with little value in an extreme result, consider 4d6-3. Add + modifiers to nuance the odds in the character’s favor.
- Ditto combat training.
- Consider using 3d6 for anything involving biological systems to take advantage of the trend toward the average. Be very aware of the impact of modifiers – which results become impossible, and which results get put on the table to replace them, and what it does to the neutral bias relative to a d20.
- When you want to take extreme results off the table but still want to preserve a lot of the evenness of results throughout the range, consider 2d10.
- You can put fumbles back on the landscape with 2d10-1 but this takes away the critical success possibility. This is recommended for a character performing a task unskilled.
- When a player indicates that near-enough-will-be-good-enough, use 2d10+#. It makes extremely good results more unlikely while increasing the likelihood of success. 3d6+# has the same effect but with a stronger bias toward the average result. This is also appropriate when time is more important than ‘pretty’.
- Whenever 210 is an option, d10+d12-1 also needs to be considered. This makes extreme results just a little more common and resists the trend to the central results a little bit more.
- When someone is being taught a skill, consider d8+d12, representing a supervisor who will gently nudge toward a satisfactory result, helping out when things get sticky. This roll makes both extremes less likely.
- When a delicate situation could abruptly swing either way, consider using 2d4+d12 instead of d20. Especially when one character is actively trying to help or hinder another. Extremely sensitive to modifiers; there’s a whole range of nuanced options to pick from.
- When you want to give players a sense that they are ‘winning’ (even if they aren’t), consider using 2d8+d6-2 instead of d20. Extreme results are more possible than on some other rolls but the overall average is higher, so success is more likely. At the same time, there is a mild push toward more average results.
- Alternatively, consider d4+d6+d12-2, which is a flatter, more evenly distributed option with greater potential for extreme results. Or d4+d8+d10-2, which is not significantly different.
- When you want the character to succeed while preserving the chance of potential failure, consider 3d8 – 3. This has an average result of 10.5, same as a d20, but that goes up by 1 for each +1 modifier to the roll. It might be easy to be too heavy-handed in this respect.
- When you want to convey to a player that they are making a stupid mistake that you don’t want to succeed for the sake of the game, use 2d6+1 (for high results desired) or 2d6+6 (for low results desired).
- When you want to convey to a player that the circumstances don’t really permit failure and probably don’t need to be rolled (but they insist or it’s an NPC doing something contrary to what the PCs would want), use 2d6+1 (for low results desired) or 2d6+6 (for high results desired).
- When constructing a table based on a probability chart drawn to your specifications, consider using d4 x d6 – d4 +1 instead and letting the roll do all the hard work.
- When you want to bring a carnival atmosphere, a sense of the absurd, into the game, use d30 +1 – d10 instead of d20. If you roll less than 1, the opponents do something monumentally stupid or something ridiculous happens to them; if you roll above 20, the shoe is on the other foot.
- When the GM thinks that there is no chance of failure but degrees of success or complications to be overcome getting to success are present, consider replacing d20 with 5d4 / d5. A gateway to roleplaying.
- When results could go either way and snowball quickly, consider using log2 (d6/3) ^ d8 +d8 +5.
If you don’t know how to do a logarithm to the base X, the trick is
logX(#) = log(#) / log(X).
For example, log(1024) is 3.0103; log(1024) to the base of 2 is 3.0103 / log(2) = 10. Which mean that 1024 is 2^10.
Absurdities that are real: 1024 = 6.0551 to the base of pi. I don’t know why you would ever need to know that, but this is the technique that lets you calculate it if you ever do.
Also consider 2d10-# when the game system states that you need a certain minimum attack bonus even to hit – it transforms ‘impossible’ into ‘unlikely’, giving your PCs a chance to survive. Works for NPCs up against PCs decked out with magical gear, too.
Replacing 3d6:
- For additional drama: consider 4d6-3. Especially to resolve skill checks in which there is significant opposition or circumstantial difficulty to overcome.
- Consider using a d20 and re-rolling any result below a threshold to describe the results of genetic modification or selective breeding.
- Consider using d8+d12 to replace 3d6 for the simulation of poisons and diseases, where some effect takes place but extreme effects are unlikely – but can be worse than on a 3d6 roll. But there are better options even than this for this circumstance.
- Consider using 2d6+d8 instead of 3d6 when the outcome is of lower importance. It has a much lower chances of an extreme result and more even chances of anything else. Modifiers are especially powerful. So trouble is more likely to happen and can be better mitigated by arranging circumstances in your favor. This encourages roleplaying AND tactical thinking.
- 2d8+d6-2 Increases the potential diversity of results, useful for situations that are on a knife-edge. Far less centrally-dominated than a 3d6 roll.
- Alternatively, consider d4+d6+d12-2, which is a flatter, more evenly distributed option with greater potential for extreme results. Or d4+d8+d10-2, which is not significantly different.
- When you want to convey to a player that they are making a stupid mistake that you don’t want to succeed for the sake of the game, use 2d6+1 (for high results desired) or 2d6+6 (for low results desired).
- When you want to convey to a player that the circumstances don’t really permit failure and probably don’t need to be rolled (but they insist or it’s an NPC doing something contrary to what the PCs would want), use 2d6+1 (for low results desired) or 2d6+6 (for high results desired).
- When constructing a table based on a probability chart drawn to your specifications, consider using d4 x d6 – d4 +4 instead and letting the roll do all the hard work.
- When you want to bring a carnival atmosphere, a sense of the absurd, into the game, use d30 +1 – d10 instead of 3d6. If you roll less than 3, the opponents do something monumentally stupid or something ridiculous happens to them; if you roll above 18, the shoe is on the other foot.
- When the GM thinks that there is no chance of failure but degrees of success or complications to be overcome getting to success are present, consider replacing 3d6 with 5d4 / d5. A gateway to roleplaying.
- When results could go either way and snowball quickly, consider using log2 (d6/3) ^ d8 +d8 +5. See the notes on d20 substitution if you don’t know how to turn a logarithm in one base (usually 10 or e) into another (for example, 2). This comes up all the time in the Hero System where +5 = twice as much. For example, adding 20 strength means you can lift 2 ^ log(20/5) times as much = 2 ^ 4 = 16 times. And 5 points of temporary stat damage means you have half as much of that stat.
When a delicate situation could abruptly swing either way, consider using 2d4+d12 instead of 3d6. Especially when one character is actively trying to help or hinder another. Extremely sensitive to modifiers; there’s a whole range of nuanced options to pick from.
And that brings to a close yet another example of “it seemed like a quick and easy post when I started”. It wasn’t – it’s been arduous and grinding, with lots of detail needing very close attention and very high levels of concentration, which were mentally exhausting – to the point where I could only do about 1/2 a die roll’s analysis in a session without pausing to recuperate and recharge.
But I think the results are worthwhile, and in some cases, fun!











































































