D&D Dice rolls (statistics)

Factual Questions
1

We are playing a game of Dungeon and Dragons(actually Pathfinder*) over email.

when a player needs to roll we use the honour system.
If a player wants to cheat, I don’t really mind
one player is having a very lucky streak

these are his dice rolls

8
20
16
12
16
13
19
20
4
19
11
17
15
14
14
18
17
14
18
17

to pass a roll you need to throw a 20 sided dice and the result needs to be higher then a certain number, differs per throw, but say 14

what are the odds of someone having this streak?
and can someone explain the math for me?

for reference these were my dice rolls

8
1
13
16
1
15
4
15
2
2
18
4
18
5
7
3
9
14
10

*Rise of the Rune lords

2

The odds of rolling numbers that are loosely within one-quarter of the die’s sides is 1:4 for each roll, so a succession like that is 0.25 * 0.25 * 0.25 and continue for the number of rolls. Pretty damned unlikely.

Everyone knows that 20-sided dice are illegal in California, right? Just BTW.

3

I’m not a mathematician, so I can’t answer the statistics question, but there is a chance he’s playing with a lucky die. In other words, some icosahedron dice are poorly manufactured and have an off-center balance.To see if this is the case, get a glass and add about 1/3 a cup of room temperature water and mix in 6 tbsps. of salt. Now have the die float in this solution. Then spin the die to see if it is consistently landing on a side.

4

That particular exact streak of 20 specific numbers? That’s easy: 1 in 20^20, which is a very large number. But that’s the wrong question to ask, because the answer is the same for any sequence of 20 rolls.

The better question to ask is what are the odds of having a result that is at least as remarkable as the one observed, or more. This one is a little tricky, because it depends on how you define “remarkable”.

For the sake of argument, I’ll consider “remarkability” based on the sum of those numbers (in other words, treat it as a single roll of 20d20). His total roll is 302, while the average of 20d20 is 210. So he’s definitely above average.

But how much above average? Well, when you’re throwing that many dice, your results will be very well approximated by a normal distribution. We already know the average of this distribution, so we just need to know the standard deviation, and we’ll know everything about the distribution. The standard deviation is the square root of a quantity called the variance, and the variance of a sum of dice rolls is just the sum of the variances of the individual rolls. A single roll of an n-sided die, meanwhile, has a variance of (n^2 - 1)/12. Thus, a single d20 roll has a variance of 399/12 or 33.25, and 20d20 has a variance of 665. The square root of 665 is about 25.79 , meaning that’s the standard deviation (we can also write this as the distribution being 210 ± 25.79). His rolls are 92 higher than the average, which is 3.57 standard deviations.

Using a cumulative probability calculator or a standard table, we find that the result has a .99982 chance of being less extreme than that, or a 0.00018 chance (1 in over 5500) of being that extreme or more.

Which still doesn’t necessarily mean anything. That’s the chance of any one streak of 20 rolls to be that good. But you’ve seen many, many streaks of 20 rolls before, and the ones that weren’t remarkable, you never bothered to ask about. How many such streaks have you seen? Well, how many people do you game with, how often do you meet, and for how long have you been meeting? If you’ve got 10 people in your group, for instance, and you’ve met every week for the past decade, then you’d expect that at some point in your gaming history, you’d get about one streak that good. Cut it down to half that many people at half that many events, and it’s still not particularly surprising.

5

I once had a Dragon Magazine that had a BASIC program to run what is called a Chi-square test to test the fairness of dice. Here is a good site that gives the test of a spreadsheet.

6

Well, the odds of him rolling that exact sequence of numbers are something like 1 in 10[sup]26[/sup] – 1 in 100 septillion. But the same thing can be said for the exact odds of rolling your sequence of numbers.

But he rolled a 14 or better in 15 out of 20 tries… I think, according to this site, Dice probability Calculator - High accuracy calculation , the probabilities of that are something like 0.02%, that is, 1 in 5000

So, within the realm of possibility, quite unlikely.

ETA: And I got ninja’d by Chronos of course!

ETA2: But it looks like we approached the problem in different ways and got very close to the same answer, which is pretty cool!

7

Seventeen die rolls is not nearly enough to give a good reading o a particular D20, but here is what I’ve found with your fiends die.


Die Face	Expected	Observed	O-E	(O-E)^2	[(O-E)^2]/E
1		0.85				-0.85	0.72	0.85
2		0.85				-0.85	0.72	0.85
3		0.85				-0.85	0.72	0.85
4		0.85				-0.85	0.72	0.85
5		0.85				-0.85	0.72	0.85
6		0.85				-0.85	0.72	0.85
7		0.85		1		0.15	0.02	0.03
8		0.85				-0.85	0.72	0.85
9		0.85				-0.85	0.72	0.85
10		0.85				-0.85	0.72	0.85
11		0.85		1		0.15	0.02	0.03
12		0.85		1		0.15	0.02	0.03
13		0.85		1		0.15	0.02	0.03
14		0.85		3		2.15	4.62	5.44
15		0.85		1		0.15	0.02	0.03
16		0.85		2		1.15	1.32	1.56
17		0.85		3		2.15	4.62	5.44
18		0.85		2		1.15	1.32	1.56
19		0.85		1		0.15	0.02	0.03
20		0.85		1		0.15	0.02	0.03
				17				21.82

At first glance, it appears to be heavily biased, but like I had said, seventeen die rolls are not nearly enough to get a good read.

for a D20, you want at least 100 rolls.

8

Maus Magill, I think you managed to corrupt your data, as the friend’s rolls are a list of 20 numbers… I think you added the 7, but left out a 4, 8, 19, and 20.

9

I’d almost say it looks like he’s rolling 2 d20s and taking the highest. A single d20 string of integers should be throwing out a median and mean around 10 but his sit at 16 and 15 respectively.

I ran 2 sets of 300 random numbers (1-20) and took the highest and I wind up with a 16/15 median/mean result.

Still, it’s only 20 rolls so maybe he’s running hot at the moment.

10

Lots of 14’s and 17’s. Possible, but I have seen cheaters stick to specific numbers.

For example, my long-former ex-friend got really lazy and it is amazing how every single die roll was a 16! Even when people were looking at the die and then looking at me to shake their head ‘no’!

I also find a lot of cheaters own the complex pattern dice that can’t be read from more than about a foot away. Or they quickly read the dice and then pick them up. One guy I played with insisted on using a d10 and d6 together, with the 1-3 on the d6 being low, 4-6 being high. Except when he really wanted to hit, and 3-6 on the d6 would be high.

Over email, it is really a lot easier to cheat. Just make up a number that sounds plausible and allows you to hit. You’re +6 to hit and you know the monster has an AC of 19? Ok, I can name any number 13 or higher. If I vary the number, maybe even ‘miss’ once or twice when it doesn’t matter, or fail that save against something that isn’t hurting me*, then no one is the wiser, right?

  • Example; They’re playing a Pathfinder character fighting a monster with the Burn special ability. Every time they’re hit by it, they need to make a reflex save or catch on fire. BUT, the group’s Wizard has cast Resist Elements(Fire) on the character, so they have Resist Fire 10 up. They can fail that save all day (or as long as the spell lasts) and suffer no consequences.

So to catch cheating players, look for the following;

  1. Illegible dice or funny dice tricks (quickly picking them up, rolling them behind their hand or away from other players.
  2. Too much consistency in which numbers come up.
  3. They always hit when they need to and save when they need to, and fail only when there’s no real consequences.
  4. Consistency in this behavior or pattern across multiple sessions.
  5. Usually, but not always accompanied by cheating in other ways.
    5a. Items they didn’t buy or find in treasure, or suddenly upgraded items.
    5b. Fudged numbers (higher attributes, saves or hit points) on their character sheet.
    5c. Constantly re-writing the character and changing skills, feats etc. This is so you never know what they have and if they know they need a feat or skill for next week’s session, they’ll have it.
11

That’s what I get for alt-tabbing between windows. I’ll come in again.

12

Okay, a bunch of people got here before me but since I tried to use math and succeeded, I’ll tell you what I found.

There are too few rolls to get a good estimate of the distribution of each face so I broke the distribution into quadrants so I could at least see if that distribution was about right.

The observations were:
Rolled from 1-5: 1 time (expected: 5 times)
Rolled from 6-10: 1 time (expected: 5 times)
Rolled from 11-15: 7 times (expected: 5 times)
Rolled from 16-20: 11 times (expected: 5 times)

I used these to prepare a Pearson’s chi square test (Pearson's chi-squared test - Wikipedia).

I came up with a Chi-squared of 14.4 and 3 degrees of freedom. According to the Chi-squared distribution here (Chi-squared distribution - Wikipedia), that outcome has less than a 1% probability of occurring by a random process. Another table showed it to be less than 0.5% probability of occurring by a random process.

Chronos is right that this sticks out at you among all the 20-sided die streaks because it was particularly noteworthy. It’s also possible his die is strongly biased. I haven’t checked how biased the poorly-manufactured dice tend to be; maybe this is the kind of distribution they produce. If you’ve seen over a million 20-sided dice rolls, a streak like this was bound to come up. But, have you really seen that many die rolls? Are you sure you would’ve noticed a streak like this if you didn’t already suspect your player of dishonesty? I’m more inclined to believe that your player is just cheating.

13

Statistically, I can’t answer your question.

Realistically, does he know that “Cursed By DM” has virtually no saving throw…? :eek:

14

Here’s what I’ve found:


Die Face	Expected	Observed	O-E	(O-E)^2	[(O-E)^2]/E
1		1				-1	1.00	1.00
2		1		1		0	0.00	0.00
3		1				-1	1.00	1.00
4		1				-1	1.00	1.00
5		1				-1	1.00	1.00
6		1				-1	1.00	1.00
7		1				-1	1.00	1.00
8		1		1		0	0.00	0.00
9		1				-1	1.00	1.00
10		1				-1	1.00	1.00
11		1		1		0	0.00	0.00
12		1		1		0	0.00	0.00
13		1		1		0	0.00	0.00
14		1		3		2	4.00	4.00
15		1		1		0	0.00	0.00
16		1		2		1	1.00	1.00
17		1		3		2	4.00	4.00
18		1		2		1	1.00	1.00
19		1		2		1	1.00	1.00
20		1		2		1	1.00	1.00
		20						20.00

It still looks a bit biased, but the point still stands that you need many more die rolls to determine bias.

15

Thanks for the feedback

I don’t mind him helping his rolls, if he is happy with having a blessed character that’s OK with me, I was just wondering what the probability would be to be that lucky.

And yes the low rolls were for a perception check someone else allready passed and a something else without consequences

16

maybe I’m reading it wrong but he didn’t roll a 2

17

Also what I think I should be asking how unlikely is it to have so many successes

I mean the chance of you throwing a 1 or higher is 100% with a D20 and 20 or higher is 5%
like this:

roll chance of or higher
1 1
2 0.95
3 0.9
4 0.85
5 0.8
6 0.75
7 0.7
8 0.65
9 0.6
10 0.55
11 0.5
12 0.45
13 0.4
14 0.35
15 0.3
16 0.25
17 0.2
18 0.15
19 0.1
20 0.05

18

That was the approach I took, willthekittensurvive.. Assuming that a 14 or higher was success, (35%), I tried to figure out the odds of succeeding at least 15 times out of 20 rolls, and got the 0.0002 (0.02%) number.

19

I remember that article! And, with a quick google, I came up with a PDF of the issue that contained that it.

20

That depends on what question you’re trying to answer, and how strong the signal is. This data set is certainly not sufficient to say that any particular face is favored on that die: There’s no single face that hits more than three times out of 20, and that’s not at all remarkable. However, when you notice that all of the overrepresented faces are high-valued ones, and all of the underrepresented faces are low-valued ones, that becomes a much stronger argument about the die as a whole. Likewise, if he rolled the die only ten times, but got a 20 every time, then that would be very strong evidence that the die was biased towards 20s.
stpauler, the two most common sources of bias in a die are an inconstant diameter or an internal bubble. An inconstant diameter means that the shape is slightly “squished” (making the two squished-together faces more likely) or “stretched” (making them less likely) in one direction. This is the reason why dice have opposite faces all adding up to the same value, so a shape defect won’t favor high or low numbers. This looks highly unlikely to be the cause of the results the OP observed, since no pair of opposite sides is particularly overrepresented.

An internal bubble will throw off the center of gravity of the die, and cause numbers close to the bubble to come up more often. This is what’s detected by the float test. It also looks unlikely to be the culprit here, though, since (assuming that his die is numbered in the usual pattern) the faces that are overrepresented are all over the surface of the die.

Of course, there are other possible sources for die bias, too, especially for a many-sided die like a d20. For instance, some of the edges might be more worn or rounded than others. But that’s much harder to test for directly.

And all of this reminds me that I never have gotten around to doing direct tests like this on my known-bad d20, which probably has both a squish and a bubble based on the statistical results (very strong bias against rolling 20s, and weaker but still definite bias against 1s).