This composite image combines a d20 extracted from dice-3563941 by Dieter Staab (plus a couple of variants rotated and color-shifted), one from dices-4804498 by Armando are (contrast & brightness enhanced), a third from rpg-468917 by Sayaka Photos (contrast & brightness enhanced), a fourth from dice-3380228 by Devin (plus a copy brightened, color-shifted and rotated), and a fifth from dice-5923500 by Renate Köppel, in front of a fractal image (abstract-art-1476001) by Patty Talavera, all from Pixabay, framing, image editing and compositing by Mike,
— all to symbolize the concept that hidden patterns may exist in the most seemingly-random of datasets.

I was reading something on Quora the other day that offered a fairly convincing argument that most people wouldn’t recognize real randomness if it bit them on the toe (in less colorful language).

Now, most GMs are not ‘most people’; we work with randomness all the time. But the more I thought about it, the more convinced I became that the majority would be just as vulnerable to the common misperception of randomness, and that understanding the difference between perception and reality could be a valuable tool.

Fake Randomness

When most people think of a random distribution of results, they actually think of an even distribution, or a scatter-plot. And, at first blush, that makes sense; each of the possible results of a d20 has an equal likelihood of occurring, and so (over many results) you would expect the number of times any given result comes up to equalize.

The problem is that most people seriously underestimate the number of results that you need in order for that to happen. For example, let’s take a string of five results: 4 – 7 – 8 – 10 – 16. Even probability means that this is just as likely to occur as another valid set of results: 10 – 10 – 10 – 10 – 10. Yet, if you were to show those strings to someone, they would have little hesitation in describing the first as random and the second as decidedly not random.

It’s a known fact that humans absolutely suck at seeing and recognizing randomness. In fact, our brains are hardwired to spot and recognize meaningful patterns taking as many shortcuts as possible in the process to speed it up. Being able to spot the tiger stalking you behind the greenery from a minimal number of glimpses holds an obvious survival benefit.

These shortcuts are what is responsible for the phenomenon of optical illusions. This is one of those subjects that I find absolutely fascinating, so I’ve dealt with it a number of times here at Campaign Mastery:

• Art-Spiration, for the Nov 2022 Blog Carnival;
• The Glow Around The Corner;
• The Power Of Blur, for the June 2021 Blog Carnival;
• A Sharp Lookout: How Much Can You Adventure?; and
• Blind Spots and False Illusions: How much can you really see?

Oh, and while I’m at it, I should probably also mention that this isn’t my first article on the subject of randomness:

• Into Each Chaos, A Little Order Must Fall;
• Pieces Of Ordinary Randomness;
• Randomness In RPGs;
• Principles of Randomness; and finally,
• The Sixes System, which has randomness and probability at the heart of the game mechanics.

So the survival strategies built into us by countless generations of kill-or-be-killed are directly at war with the ability to distinguish the absence of pattern, to such an extent that the mind will try and invent a pattern when none exists.

This reality is responsible for all sorts of things, from Optical Illusions to Eyewitness Contamination to Social interaction during Jury deliberations to Conspiracy Theories and, perhaps, to an even broader application to paranoia itself.

The Truth About Randomness

In reality, then, true randomness doesn’t look anything like an even distribution. So what does it look like, then? And what light can be shed on the number of rolls that you need before a reasonable level of uniformity of result (as perceived and defined by a layman) can be observed?

These are far more complicated questions than it might first appear, so they will take some time to answer.

Distribution Of Results

To provide a real-world analysis, I rolled and documented 448 d20s. The graph analyses the results.

Why 448? Well, it’s more than 400, and 400 divided by 20 is 20; my instincts were that having the average tally of any given result be so high would be enough to show just how uniform the occurrence of any given result would be.

At the top, you can see the actual distribution of results – 16 times, I got a 1, 13 times I got a 20, and so on. A whopping 36 times, I rolled a 16! Surely, that means that the rolls weren’t truly random? That’s one result with a little over half the expected tally, and another with almost double it!

Well, when you calculate the average result, you get 10.4 – which is a smidgen below the theoretical average of 10.5. So maybe we need to look a little more deeply into these results.

Some Analysis

There were 448 rolls, so even distribution would be 22.4 occurrences of each result. What has actually been observed ranges from the low of 13 to the high of 36. Those are differences of -9.4 and +13.6, or -42% to +60.714%. So what can be said is that 448 rolls yielded an average of 22.4±61% occurrences per result.

What’s more, it’s reasonable to expect that this margin of error would probably halve each time you doubled the number of rolls. So, at 996 rolls, we have a probable error margin of ±30.05%. Let’s round those to 1000 and ±30%, for convenience.

Double again, and we get 2000 rolls and ±15%. Again yields 4000 rolls and ±7.5%. and, once again to get 8000 rolls and ±3.75% – finally a margin of error that is smaller than the range of the results (5% vs 3.75%).

UPDATE

A comment to a repost of this article on another site has pointed out that the ‘reasonable to assume’ is actually incorrect. To halve the error margin actually requires four times as many tests, to reduce it to a quarter requires sixteen times, and so on.

Which means that to get a probable error margin of ±30% requires roughly 2000 rolls, to get that down to ±15% requires 8000 rolls, ±7.5% needs 32,000 rolls, and ±3.75% needs 128,000 rolls.

I don’t think this makes any material difference to the remainder of the article, but bear it in mind. Individual results are far more smeared all over the map, more chaotic, than I thought they were.

Huge thanks to Andrew for passing on the feedback. Much appreciated!

More Graphical Analysis

However, die rolls are notoriously non-linear in their probabilities, which I’m at pains to point out in my analysis of the mechanics of the Sixes System. The normal pattern when it comes to a standard distribution is a core of very flat probability, in which variations are commonly observed, surrounded by a region on the curve in which the number of results rises or falls at a steep angle, surrounded by a plateau of very low probability results.

My standby tool for evaluating such large numbers of die rolls is Anydice, but when I went there, I found that this many dice went beyond it’s accepted limits. Instead, I had to drop the number of dice to 112 – so it’s not going to be directly relevant. But the principles will still be the same.



Base curves plotted with Anydice, refer link above.

If you’re talking about 448 die rolls, the central pyramid is only going to be narrower. At the same time, there are so many results with virtually zero chance individually that one or two anomalous results would not be surprising.

But this is all misleading, because we’re talking about 448 individual rolls, not one roll that compounds 448 dice. It is this difference that explains, and causes, the erroneous interpretation of randomness; on any individual roll, the chance of any given result – from one to twenty – is 5%.

The chance of two of them is 5% of 5%, or 0.25%. The chance of three is 5% of 0.25%, or 0.0125%. The chance of 4 is 0.000625%.

448 individual rolls is 224 pairs of rolls, so applying these percentages, we get:

• 448 rolls at 5% = 22.4 ones, 22.4 20s, and so on.
• 224 rolls at 0.25% = 0.56 pairs of 1s, 0.56 pairs of 20s, and so on.
• 112 rolls at 0.0125% = 0.014 triplets of 1’s, 0.014 triples of 2’s, etc.
• 56 rolls at 0.000625% = 0.00035 strings of 4 ones, strings of four 20s, etc.

Hmm – I’m not sure this adds much to the conversation.

Maybe it’s the whole concept of aggregating die rolls that’s leading us astray. So let’s contemplate a way of thinking about the sequences of results, translating them into some sort of graphical display.

Sequential Results

Clearly, mapping each actual result onto a single space on a grid is going to be fairly useless. What’s needed is some way of consolidating individual results into shorter strings of results.

The method that I decided to use, after some thought, was to roll d20s and map them onto a single row of a horizontal grid until a result came up that matched a result that had already been rolled; since this would not fit on the existing row (that space was already occupied), it would force the shift to a new row. I further broke them up into four sets of results to make the graph more convenient.

With a set number of rows to fill, the decision was to keep rolling until a result came up that ‘fell off the bottom’, signaling the end of the run. That, of course, explains the reason for the odd number (448) rolls.


Length Of Result Sequences

I also thought it important to analyze the theoretical length of the resulting strings – I didn’t want them to be too short, or too long. Because it made the math easier, I did this theory as a chance out of 400.

• Length 1: 1 in chance in 20 = 20 / 400 (by definition).
• Length 2: 19 attempts at 1 chance in 20 of matching 2 results = 38 / 400; subtotal 58.
• Length 3: 18 attempts at 1 chance in 20 of matching 3 results = 54 / 400; subtotal 112.
• Length 4: 17 attempts at 1 chance in 20 of matching 4 results = 68 / 400; subtotal 180.
• Length 5: 16 attempts at 1 chance in 20 of matching 5 results = 80 / 400; subtotal 260.
• Length 6: 15 attempts at 1 chance in 20 of matching 6 results = 90 / 400; subtotal 350.
• Length 7: 14 attempts at 1 chance in 20 of matching 7 results = 98 / 400; subtotal 448.

So the average length of a string will be 6-7. Which means, out of 20, that about 2/3 of each row will be empty space. That seems like it will be enough.



True Randomness

This is the result, prettied up a bit, and right away you can see that true randomness is lumpy, coming in clumps. There are huge voids, like the one just below the right-hand top corner, and a smaller one just above the center. There are long strings of sequential results 2, 3, 4, and even 5 long. And there are a number of vertical bars that indicate the same number recurring time after time.

Below is an animated graphic showing a random walk with 25000 steps. It shows the same clumps and voids as my d20 results, and for exactly the same reason: randomness is not uniform in results, only in the likelihood of results (and sometimes not even then).. .

The misinterpretation has been responsible for a number of superstitions and fallacies that remain commonplace today.

The fallacy that a result is ‘due’, for example. If you are flipping a coin, the coin has no magic memory that makes a given result more or less likely – it doesn’t matter if you have just gotten 5 heads in a row, there is still a 50-50 chance of getting a head with your next coin-flip.

The fallacy that a past observed trend resulting from true randomness will persist, or be reversed, gives rise to the superstition that some numbers are more or less likely to result. To take an example from the die rolls that I have tracked in preparing this article, the number of results of “16” doesn’t mean that I’ll keep rolling a disproportionately high or a disproportionately low number of 16s in the near future.

In exactly the same way, the relative lack of twenties doesn’t mean that I will roll extra 20s to make up the shortfall anytime soon; it might happen, and will probably happen eventually, but it could be in 20 rolls or 2000.

Animated random walk with 25000 steps by Laszlo Nemeth (anglicized credit), CC0, via Wikimedia Commons

No surprise – non-random digit distributions

The distinction between perceived randomness and true randomness might have surprised some. It might even have surprised me, but as soon as I saw it, my mind connected it to another phenomenon: non-random digit distribution.

If you invent a supposedly random series of multi-digit numbers, there will be a preponderance of threes and sevens in the digits. People tend to avoid even numbers, fives, nines, and zeros when inventing numbers because they perceive these as ‘less random’ than they should be.

This is one of the obvious consequences of the difference between true randomness and perceived randomness.

Note that you have to exclude leading digits in such analyses, because of Benford’s Law.

Benford’s Law

The leading numbers of any long series of numbers is going to be disproportionately low. This makes total sense when you think about it for a minute.

• In the numbers 1 to 20, eleven of the results will start with a 1, and two of them with a 2.
• In the numbers 1 to 200, one hundred and eleven results will start with a 1, and twelve with a 2.
• In the numbers 1900 to 2023, all but 24 of the numbers will start with a 1, and the rest will start with a 2. This is a completely not-random distribution.

Benford’s law, “also known as the Newcomb-Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.” –Wikipedia

While it’s obvious why Benford’s Law applies to some data sets, in others it is simply an observed fact that resists simple explanations. To test for randomness of digit distribution, you therefore have to exclude the leading digits.

No surprise – chits

Random sequences are a different thing to a set of random numbers or random die rolls.

For example, consider what happens if you are drawing chits numbered from one to twenty without replacing them. The sequence of results will be random, but each draw is not an independent variable, because you cannot draw a result that has already been observed. Every result (assuming you continue until all the chits have been drawn) will occur, but the sequence will be random.

This actually sheds light on both true randomness and perceived randomness.

Public Domain Image supplied by Wikimedia Commons.

The Perceived Randomness Significance

When people think of what they perceive to be random distribution, the result is not unlike drawing chits from a bag for each coordinate on a chart. The accompanying illustration is a 200×200 grid containing 20,000 dark points (out of 40000) – so an even distribution of black and white. This is a plot of noise, basically, but it’s still what most people perceive as randomness.

If you keep adding dark pixels at random, sooner or later, all the white will be gone. Drawing chits from a bag instead of using a random number generator simply cuts out any intervening span by ensuring that you are selecting from the current population of white pixels only.

The True Randomness Significance

Below is another random walk pattern that was generated using pseudo-random numbers. At each step, the black could move into any of the nine cells surrounding it (which includes back the way it came, or staying where it was). After 2,000,000 steps through the 40,000×40,000 grid, the process was halted and the current state captured in the image which was then cropped.



Purpy Pupple, CC BY-SA 3.0, via Wikimedia Commons.

I chose this image because it clearly shows the clumps and voids of true randomness. Which is ironic, because it’s not actually true randomness being displayed – results are constrained to be right next to the current pixel. This is not the same constraint as occurs when drawing chits from a bag, but the effect is similar, in that each point of white is simply a coordinate that hasn’t been ‘explored’ yet. Once again, if you continue generating steps, and exclude those results that take the ‘point’ out of the 40,000 pixel square space, the entire space will eventually become filled with black.

What’s more important – randomness or the Perception of Randomness?

All this leads to the inevitable question, which one – perceived randomness or true randomness – should a GM aim to use in his games?

After careful consideration, I don’t think there is any one right answer to this question; depending on the circumstances, either could be correct.

Repeat Exclusions

If what you want to do is to generate a sequence of some kind, but no choice is to be repeated, the chit-draw approach is the best choice. For example, if the intent is to determine the sequence in which the PCs interact with the individuals listed as patrons in a bar, this is the better approach.

True Randomness

Only if you don’t intend to have all possible outcomes occur should you consider alternatives. If the intent is simply to flag a course of action or plot, or to present a limited number of the possible outcomes, a die roll is the better approach.

For example, if you are generating a list of ‘serial numbers’, rolling d10s for each digit (rolling multiple dice and reading them left to right) achieves the randomness that should be there – though it may be necessary to insert a leading digit generated with a smaller die or even a fixed set of leading digits.

…with overrides

The basic technique of rolling for digits (or whatever) will serve many but not all needs. For example, if you want to generate a simulated fraudulent sequence or list of sequences, you might roll a d6 as well as a d10 for each trailing digit – on a 1, you override whatever is showing on the d10 make the digit a 3, on a 2, you override the d10 to add a 7, on anything else you ignore the d6.

If the results look too obvious, you could replace the d6 with a d8 or even a d12.

Perceived Randomness

If, however, a list is to be presented to the players, it may be more useful to aim for Perceived Randomness and not true randomness. Perceived randomness, for example, will feel ‘fairer’ than the real thing (even though it probably isn’t).

It all comes down to the keyword, ‘Perceived’. If perceptions of the results are important, then willfully making choices to create the sense of randomness is too important to leave to the chance of true randomness – better to do something that will ‘feel’ right.

A good example is weather – if you were to create a random weather generator, then it would be very easy to set it up to be true random, but this would be quite unrealistic. Seasonal variations or modifiers should apply, obviously, and so should yesterday’s weather. But it would be better to create such a table or subsystem and then use it only as a guide so that you can ensure that the weather ‘feels’ random, even if it is not.

Randomness is a lot more complicated than most people realize, but an awareness of the differences between reality and perception can be a valuable asset, and manipulation of game or plot variables to create the desired impression can be a useful tool.

One Final Example: An Adventure

Let’s contemplate a mystery adventure that takes these thoughts into account. First, decide what the mystery is, and what the solution to the mystery is going to be. Next, list a series of clues that will lead the PCs to the point where they have everything they need to solve the case. Then, create a mini-adventure whose reward is one of these clues.

Some of the clues can nullify or reveal lies that are initially presented to the PCs – remember the axiom that every suspect / witness in a mystery should leave something important out or lie about something, and if the latter, they should have a strong motivation for the deceit.

Throw in a concluding mini-adventure in which the PCs deduce the identity and motive of the criminal and act on that knowledge, and you have a complete plotline. Oh, and the criminal should do their best to look both innocent and to steer suspicion on someone else, throwing out at least one red herring, as should anyone else with a prejudice or a penchant to indulge in conspiracy theories.

Here’s the thing: you could present these clue-reward mini-adventures in sequence, or in true randomness, but the better choice would be to harness the perception of randomness so that one clue can logically lead to the next.

Let’s work some numbers:

Confrontation & Resolution: 30-60 minutes.
Post-resolution / wrap-up: 15 minutes.
Mini-adventures:

Number of clues: 4.
Number of lies / distortions to be revealed as clues: 6.
Number of red herrings: 2
Number of clues to the nature of the red herrings: 2
Total number of mini-adventures: 14.
Average length of mini-adventures: 15 minutes each.

Subtotal of mini-adventures: 210 minutes = 3 hr 30 min.
Initial mystery: 30 minutes.
Preface / introduction / preliminaries: 20 minutes
Total time: 30-60 + 15 + 3 hrs 30 + 20 = 4 hrs 35 to 5 hrs 5 min.

All told, a quite reasonable adventure. If you were to increase the length of the mini-adventures (15 minutes average is a little on the short side) to 20 minutes (average), you would add an additional hour and ten minutes to the total. Which means that adding a further 10 minutes to each to bring the average up to 30 minutes each would take the estimated playing time for the adventure to 8 hrs 5 min to 8 hrs 35 min – a big day’s play, or two (more moderate) game sessions.
This is probably a better target length, simply because there’s quite a lot to happen in each!

Of course, knowing the average length allows you to design these ‘to order’ – setbacks, complications, character interactions, and so on.

You could run those 14 mini-adventures in a random sequence, or simply list the focal point and leave it up to the PCs which one to pursue next. But it would make more sense to map out two or three different sequences of arranging the mini-adventures that is a combination of logic and perceived randomness.

Randomness can be your friend, if you’re willing to work with it.
Why? Because the ‘clumping’ of true randomness can feel forced and not random at all.