On the Even Distribution of Dice Rolls

An APBA/Stat Preamble

As anyone that speaks to me for more than 12 minutes typically finds out, I’m hopelessly addicted to a game called APBA (or “geek baseball” among anyone who doesn’t play). As a refresher, the core concepts of the game are to take a full season’s stats, create player cards based on said stats, and then roll dice to simulate games. From a baseball fan standpoint it’s a great game, but from a statistical standpoint it’s fascinating - we’re effectively able to use random events (dice rolls) to simulate events based on previous real results.

One of the core components of the game is rolling two six sided dice to come up with a combination of 36 results ranging from 11-66. The results of individual dice are not added, just concatenated in a specific order. In “face to face” play, a red and white die are used with the red die providing the first number and white providing the second.

As any stat geek will know, an even distribution of these results is critical to simulating previous results. Skewed results will not only impact the results of the game, but could give players an unfair competitive advantage. In a 12 team league like I play in, this could impact results at worst and cause drama and accusations at worst.

Some Contextual Whining

In my current APBA league it’s widely acknowledged that I’ve lucked into having a very good team this year. Talk to any baseball fan about an outfield of Stanton, McCutchen and Trout, Abreu at 1B and a variety of very strong pitchers and they’d be impressed. Talk to APBA players about it and they’ll grin in jealousy.

Oddly, this roster has produced surprisingly poor results for me this season. Their expected stats are notably below what they should be. In a game where sample sizes are readily available, these guys all had a surprisingly poor first half of the season. Typically this is blamed on “cold dice” - just pure bad luck of the dice rolls that are widely accepted as a part of the game.

At this point I’m just whining...my really good team isn’t hitting as many homeruns as they should. Waaaaahhhhhhh. But I started noticing a few other things:

• My opponents seemed to be hitting a lower than normal number of homeruns (in APBA terms, rolls of 11s and 66s seemed to be down)
• Everyone seemed to be hitting a lot more “secondary power numbers”, singles and walks (APBA terms: rolling lots of 33s, 44s, 45s, etc)
• The dice distributions just *seemed to be off*. (APBA terms: I had no proof other than a gut feeling)

After half a season things just weren’t sitting well with me. Something seemed off, but I had no real evidence to point to.

As Quick As Possible: Dice Rolling

In the old days our league members all lived in or around Washington, DC and we all played face-to-face games. That’s right, we’d schedule times, drive to each other’s houses and roll dice/talk about baseball/eat pizza for 3 hours and have a grand old time. Unfortunately lives happen, people move and years later we have people scattered across the entire US, so face-to-face games are not possible.

Luckily, the internet exists! Almost everyone knows about AIM and chat windows. Luckily AIM has a dice rolling function that can be used! Unfortunately...

1. Fewer and fewer AIM clients now support this functionality
2. AIM Express has been decommissioned, meaning one less client supports it
3. Mac fan boys like me are nearly SOL for using AIM to roll (note - it turns out that OS X Messenger can somewhat support rolling)

Worry not though, the internet still exists! As do some (seemingly) great dice rolling sites. This is where things get a bit curious.

The (Worrisome) Data

The goal of this post is not to knock down a pet project, so I’ll not call out the site I’ve been using this year, but I will point out the dice results I received from some simulations. Take a look at the below table.

(Update: Udo from Rolz.org has requested I point out this is their site, so here it is. Many thanks to him for helping with the analysis!)

Roll	# Occurrences	Roll	# Occurrences	Roll	# Occurrences
11	13	31	22	51	22
12	25	32	43	52	47
13	29	33	52	53	48
14	36	34	46	54	35
15	24	35	50	55	52
16	8	36	20	56	21
21	28	41	27	61	17
22	51	42	47	62	27
23	49	43	47	63	18
24	49	44	53	64	23
25	45	45	43	65	22
26	25	46	25	66	11

This is based on 1200 rolls (n=1200) so you’d expect the number of occurrences to be right around 33 (1200/36 = 33.3333…). A quick look shows that the number of occurrences is all over the place though, ranging from 8 on the low end to 53 on the high end. Naturally a lot of factors play into the number of rolls, and it’s not uncommon to see such a huge range. Had I paid enough attention in stats my senior year, I’d begin discussing variance and standard deviation, but there’s still a bit more to study up on before I can do that :)

Needless to say, something smells a bit fishy here. It could come down to the randomness of the dice, but let me present the results in a different way...ordered by number of occurrences:

Roll	# Occurrences	Roll	# Occurrences	Roll	# Occurrences
16	8	12	25	34	46
66	11	26	25	42	47
11	13	46	25	43	47
61	17	41	27	52	47
63	18	62	27	53	48
36	20	21	28	23	49
56	21	13	29	24	49
31	22	54	35	35	50
51	22	14	36	22	51
65	22	32	43	33	52
64	23	45	43	55	52
15	24	25	45	44	53

Notice anything odd? I certainly do.

• Let’s start first with dice rolls in the rightmost column. Not a single one includes a 1 or a 6. That seems rather odd.
• Let’s jump to the leftmost column. Ok, that’s really odd...EVERY single one involves a 1 or a 6 AND the number of occurrences is at best HALF of those in the right hand column.
• And now the middle column. Remember the expected number of occurrences for any one roll should be right around 33? Well the only rolls with either a 1 or a 6 in them that are anywhere near that are 14 (36 occurrences) and 13 (29 occurrences).

I could likely go on if I wanted, but the case is pretty clear that this dice roller does NOT like 1s or 6s. Who knows the reasoning (I can almost guarantee it’s not nefarious), but it’s safe to say that anything resembling a completely random and somewhat even distribution will not result from this particular dice roller.

The (Presumed) Gold Standard

The entire goal of an online dice roller is simple: provide results that resemble what you’d expect from real, physical dice. In the real world we’d expect a uniform distribution of results (remember, we’re not adding the totals, so rolling a 43 is different than rolling a 34). In other words, we’d expect 11 to show up as many times as 12, as 13…,as 65, as 66. Naturally chance comes in to play, so this won’t always happen, but it should be somewhat close.

To date, our league just assumed the gold standard for dice rolling was AIM. It was THE dice roller to use as far back as 1997 (save your comments ICQ, IRC, etc fans...AIM was the first chat client used on massive scale), supported dice rolling for a long, long time and never seemed to raise eyebrows.

Unfortunately rolling a lot of dice at once is hard on AIM - they throttle the number of requests (7 req/10 secs from what I can tell), so it’s a tedious exercise of data entry. Even with that being said, I was able to get n=360 rolls in (expected number of occurrences = 10) with the following results:

Roll	# Occurrences	Roll	# Occurrences	Roll	# Occurrences
11	13	31	15	51	8
12	6	32	7	52	17
13	12	33	9	53	10
14	12	34	10	54	10
15	10	35	9	55	8
16	8	36	6	56	7
21	9	41	9	61	13
22	10	42	14	62	15
23	15	43	7	63	5
24	12	44	7	64	9
25	12	45	13	65	7
26	12	46	9	66	5

Overall it’s kind of boring. There are a few outliers (17 rolls of 52, 6 rolls of 12), but those can be attributed to a smaller sample size. Yet again, remember some more from that stats class in college would be helpful here.

If we’re to sort by number of occurrences we get:

Roll	# Occurrences	Roll	# Occurrences	Roll	# Occurrences
63	5	21	9	14	12
66	5	33	9	24	12
12	6	35	9	25	12
36	6	41	9	26	12
32	7	46	9	11	13
43	7	64	9	45	13
44	7	15	10	61	13
56	7	22	10	42	14
65	7	34	10	23	15
16	8	53	10	31	15
51	8	54	10	62	15
55	8	13	12	52	17

Again, a bit inconclusive and random, but that’s what you want. It may look a little weird due to the sample size, but there appears to be no bias against any one number or role in particular. Based simply on gut feel and a long history with AIM alone, I’m not opposed to continue to use it.

An Alternative (Maybe)?

Being that I’m a tech consultant by training, it’s been pounded into my mind that it’s useless to whine about an issue without proposing a solution. This one is particularly tricky for a few reasons though:

1. AIM doesn’t work for certain users wanting to roll dice (particularly me)
2. The above rolling site is obviously not ideal for simulating real dice rolls
3. Findings like the above will naturally make anyone concerned about similar technologies.
4. Other members of my baseball league may challenge new technologies or change of any kind. It’s simple human nature and very understandable in this case.

So what’s one to do? Easy - pause in frustration and browse reddit. It’s what I do on an hourly basis anyway, so I might as well try it here.

Luckily for me:

1. Reddit has a great community
2. Other people need dice rollers
3. Google has awesome engineers who are Mac fan boys and probably got irritated when AIM Express got shut down and left them without a dice roller

The result? Google Hangouts has a random dice roller!!! That’s right, simply type /roll 2d6 in a Hangout window and a result similar to “Chad rolls 2d6 and gets 7. [ 2 5 ]” will pop up. Now the real question...will it burn when I roll as much as that other site did? To the analysis we go!

The (Hopeful) Data

Let’s start where we did with the previous data dump...a list of occurrences by dice rolls. One small note though - due to tester (aka me) miscounting, the sample size and expected number of occurrences will be a bit different. In this case it will be n=1080 with an expected number of occurrences at 30 on the dot. Now to the table:

Roll	# Occurrences	Roll	# Occurrences	Roll	# Occurrences
11	22	31	26	51	38
12	38	32	25	52	22
13	30	33	30	53	24
14	26	34	26	54	42
15	44	35	39	55	27
16	46	36	41	56	24
21	22	41	43	61	30
22	33	42	20	62	24
23	20	43	29	63	22
24	29	44	37	64	29
25	31	45	30	65	28
26	25	46	26	66	32

A few things immediately jump out here:

1. The range of occurrences changes dramatically with the low being 20 and the high being 46. Compare this to the range of 8 to 53 on the previous roller and it’s quite an improvement.
2. The bias against 1s and 6s seems to be gone. In fact, 16 was the most rolled number this time around.
3. The “gut feel” is a bit better about this table. Sure there are extremes, but most of the number of occurrences *seem* to be in the right range. Again, remembering a bit about variance and standard deviation would probably help my argument a lot here.

Now let’s look at the numbers as we did in the previous data dump - sorted by occurrences.

Roll	# Occurrences	Roll	# Occurrences	Roll	# Occurrences
23	20	31	26	25	31
42	20	34	26	66	32
11	22	46	26	22	33
21	22	55	27	44	37
52	22	65	28	12	38
63	22	24	29	51	38
53	24	43	29	35	39
56	24	64	29	36	41
62	24	13	30	54	42
26	25	33	30	41	43
32	25	45	30	15	44
14	26	61	30	16	46

See any patterns here? Seriously - I’m asking because I don’t. Sure there is a range with a few outliers on both ends, but nothing that a little math probably couldn’t explain. And there’s no obvious bias like before. Naturally I could be missing something, but in terms of gut instincts, this type of distribution just *seems* better.

The Fallout

This data is certainly some combination of interesting, compelling and daunting. Over the next few days I anticipate concern/confusion among league members, an unknown response when I attempt to helping provide some feedback to the person who’s turned a pet project into a nice little rolling site, and hope for an adoption of Google Hangout in rolling so I don’t have to go out and buy a Windows machine.

What is sure is this...as new rolling technologies inevitably must be adopted in the future, there is a 100% chance that I’ll do some basic distribution analysis to determine if there skewed in a way that could be considered flawed.

TL;DR: Use AIM or Google Hangout to roll dice, they seem legit.