Thursday, May 31, 2012

Random Dice?

Although it looks like I've significantly decreased my blogging, the truth is that I do blog posts relatively often. Unfortunately, they're usually Warhammer 40,000 related and thus have a limited appeal. Ironically, although this blog gets more hits, it doesn't get many actual visitors. The hits are because a lot of my monster images show up on a Google image search. The Atomic Spud's Warhammer 40K Blog has a lot fewer hits, but more visitors who are actually there to read the content.

Anyway, the following post originally appeared on my Warhammer 40,000 blog and may be amusing to those who aren't usually interested in the Grim Dark Future. In a way, it ties into my previous post, "How To Care for Introverts", since the activity it describes (i.e., testing dice to see if they're truly random) seems like one a typical introvert might take part in.

For the uninitiated, here's a quick explanation of some of the vocabulary. A save is a dice roll that might save a model (usually some sort of soldier) from being killed and taken off the table. The save can be due to his armor, his position in cover, or some other special rule. When I say that a save is a "3+", for example, that means that the model is saved by a dice roll of 3 or higher. The term "d6" refers to a standard die; i.e., one with six sides. Warhammer 40K is played solely with d6s, which determine whether a model hits its target, whether it's wounded by a weapon, whether it passes a save, etc. Many roll playing games (RPGs) use d6s as well as other types, such as twenty sided dice (d20s). "Warp-tainted" is a 40K term that means that something has been corrupted by the chaotic energy of the Warp, which is essentially a psychically charged version of the "hyperspace" that shows up in so many sci-fi stories. "GW" is "Games Workshop", the company that produces 40K.

Uncovering Warp-Tainted Dice
I've mentioned before that I've had some pretty horrible luck with my dice. In the past few months I've failed four out of about seven 3+ cover saves, failed five out of six 3+ armor saves immediately after failing five out of six 3+ cover saves, rolled absurd numbers of ones and twos during two out of three games, and, in last week's game, rolled multiple snake eyes, failed to wound three out of five Orks after hitting them with a strength 10 Demolisher Cannon, and failed three out of five artificer armor [i.e., 2+] saves. As a Space Marine player, I spend a lot of points for awesome armor saves and a high ballistic skill, all of which can be completely neutralized by an excessive number of ones and twos. Space Marine armies simply aren't large enough to tolerate that kind of rolling.

Being paranoid doesn't mean
that they aren't out to get you
At first I was willing to believe that I simply perceived the rolls as being skewed due to the human tendency to selectively remember extraordinary events. After a while, I started testing my dice, repeatedly rolling each die in my box of 36 dusty blue and copper Chessex dice (yes, I chose the dice to match my army). I averaged the results for each die respectively and found that they usually produced a result close to the expected 3.5. This seemed to confirm that I was selectively remembering the bad rolls.

After last week's game, I came home convinced that something was awry with many of my dice. After a Google search, I came across several blogs and websites claiming that mass-produced dice such as those made by GW or Chessex can be irregular enough to produce skewed results. This is particularly true of the d20s that various RPGs use. Many of these unbalanced dice will tend to produce huge numbers of opposites; a die that produces too many ones or twos can often produce too many fives and sixes as well. I soon realized that my initial tests were worthless; an unbalanced die that favors mostly ones and sixes will tend to have an average roll of 3.5. As a Space Marine player who usually needs only 3+ and 4+ to do what I want, I don't want dice that will produce an unusual number of fives and sixes (which are overkill) but also give me a ton of ones and twos (which represent failed armor saves and missed shots). Instead of making the expected 66.7% of my armor saves or to hit rolls, extremely skewed dice could give me an armor save and to hit rate of nearly 50%.

I eventually came across this D&D blog, which described using a Pearson's chi-square test to validate whether or not a d6 or a d20 is fair. This test has the advantage of detecting when individual numbers come up too often or not often enough rather than depending on averaging results. The equation for a d6 is:

Σ[(O-E)^2/E] > 11.070

Where "O" is observed frequency and "E" is expected frequency. The value of 11.070 is the chi-square value for a system with five degrees of freedom; i.e., the number of possible results (six for a d6) minus 1. The equation essentially says that if the result of the equation is greater than the chi-square value, then the claim that the die is fair is probably false.

To use this equation you roll a die a certain number of times and record how many times each side shows up. For each of the six possible die results, you subtract the expected frequency that the side should appear from the observed number of times that the side actually showed up. You divide that value by the expected frequency and then add up the resulting numbers from each of the six possible results. If the sum is greater than 11.070, it is statistically probable that the die is not fair. Although 30 rolls (with an expected frequency of 5 hits for each side) is considered to be a minimum, the same D&D blog noted that a 30 roll test is likely to miss slightly unbalanced dice. A better test would use more than 100 rolls per die.

I decided to apply this test to my own dice and started by placing them in a grid and assigning each die a letter and number designation from A1 to F6. I'm crazy, but not crazy enough to roll all 36 of my dice over 100 times, so I started with just 30 rolls per die. I put the data into a spreadsheet that executed the equation and also calculated how often the die gave a result of 2+ (e.g., a Terminator armor save, the usual value needed to kill infantry with anti-tank weapons), of 3+ (e.g., a power armor save, the value needed for a Marine to hit), and of 4+ (e.g., a Scout armor save, the value to wound with most sniper weapons). I used the conditional formatting feature to automatically color code results that I thought were suspicious. After crunching the numbers, I found that none of my dice failed after only 30 rolls, but that several were close. The blog's warning about a 30 roll test being insufficient became apparent when one die failed to roll a single 1 but was still able to pass the test (although just barely).

Those dice that were close to failing or that I found suspicious (18 out of the 36) were given an additional 60 rolls (90 total rolls with an expected frequency of 15 hits per side). The additional results showed that a few of the 18 dice merely had a bad series of rolls earlier and that they were relatively well balanced. However, four dice outright failed the chi-square test. The worst offender would roll a one more than a quarter of the time (27.78%) and would make a 3+ armor save only 53.33% of the time versus the expected 66.67%. Another die would only roll a 4+ 33.33% of the time rather than the expected 50%. Of the four failures, only one rolled unusually high.

I would call this one "The Widowmaker" if Astartes had widows

Some would call this one "lucky". I retired it, too.

After more than 2,000 individual dice rolls, I decided to "retire" the four failed dice as well as five others that gave results greater than 6 (a little more than half of the chi-square value). And yes, I also removed the high rolling die, which had the second highest degree of bias. In short, I removed 1/4 of all my dice from my set based on this test. Fortunately I play Space Marines, so I don't really need all 36 dice anyway. Oddly enough, I can't bring myself to throw out the biased dice. They're segregated in their own little Ziploc bag labeled with dire warnings.

I would recommend the above test for any player who thinks his dice are giving him a raw deal. I was somewhat relieved to find that, although a lot of my bad luck may be a matter of perception, a significant amount of it may be due to biased dice. And if your dice don't turn out to be biased, then at least you have mathematical proof that it's all in your head.

I guess the question is, what can a tabletop gamer do about the problem? There are a variety of dice companies that promise balanced dice, but they charge as much for five dice as Chessex does for 36. Plus, Chessex sells a greater variety of colors and styles that are great for those players who want to match their dice to their army. I've simply decided to order a second box of Chessex dice (blue and white to match Ultramarine veterans) with the intention of performing another set of chi-square tests to weed out the biased ones. The test takes me about as long to finish as two long games of 40K. If that saves me from any more dice-sabotaged games, then it was worth it.

Wednesday, May 9, 2012

"How To Care for Introverts"

I found a copy of this poster on Facebook last month:

I was amazed at how accurately this list describes myself and, to a lesser extent, several of my friends. Supposedly, around 75% of Americans are extroverts while those of us who are introverts are in the minority. Unfortunately, that means that a lot of people don't understand our personalities or realize that what's normal and enjoyable for them can be a veritable nightmare for us.

Numbers 1 and 6 have required a lot of patience on the part of my wife. Although I love my family more than anything, my wife and kids have learned that daddy's sanity is often dependent on him having his "daddy time"; i.e., a period of time in which I can sit in a quiet room listening to music through my ubiquitous noise-blocking earbuds. (I've since come to realize that the reason I constantly listen to music is so that I can control the sound around me. Without that control, random noise becomes horribly distracting.) My wife learned about number 6 long ago when she realized that I have extreme difficulty with spontaneity or sudden changes to my routine or my plans.

Numbers 11 and 12 reflect on an introvert's aversion to large social gatherings. For example, I don't know how many people tried to get me to 'get out and have some fun' at a dance or some other activity when I was young. I remember a young woman who was frequently guilty of this and who was able to drag me to the occasional dance. The only thing that made those activities tolerable was the fact that I liked her. That young woman, who became Bride of Atomic Spud just over 10 years ago, eventually learned number 12: "don't try to remake [introverts] into extroverts". The reason why we don't enjoy social events with large groups of people isn't because we haven't been exposed to them, it's because we simply don't like them. It's in our personality to dislike large social events, just as it is in the personality of many people to enjoy them.

Wikipedia seems to have a pretty good definition of introversion and extroversion that tends to agree with the poster:
Extraverts tend to enjoy human interactions and to be enthusiastic, talkative, assertive, and gregarious. They take pleasure in activities that involve large social gatherings, such as parties, community activities, public demonstrations, and business or political groups. Politics, teaching, sales, managing and brokering are fields that favor extroversion. An extroverted person is likely to enjoy time spent with people and find less reward in time spent alone. They tend to be energized when around other people, and they are more prone to boredom when they are by themselves.
On the other side are introverts:
[Introverts] often take pleasure in solitary activities such as reading, writing, using computers, hiking and fishing. The archetypal artist, writer, sculptor, engineer, composer, and inventor are all highly introverted. An introvert is likely to enjoy time spent alone and find less reward in time spent with large groups of people, though he or she may enjoy interactions with close friends. [...] They prefer to concentrate on a single activity at a time and like to observe situations before they participate, especially observed in developing children and adolescents. They are more analytical before speaking. Introverts are easily overwhelmed by too much stimulation from social gatherings and engagement, introversion having even been defined by some in terms of a preference for a quiet, more minimally stimulating environment.
Although I've never been comfortable in social gatherings, it seems to have become worse since college. To say that I'm "easily overwhelmed by too much stimulation from social gatherings and engagement" is an understatement. A couple years ago I found myself at a neighborhood Christmas party being held by some friends. I was with my wife and I knew nearly everybody in the house. Although I initially enjoyed talking with our friends, after about an hour I became quiet. Every voice and noise in the house seemed to become amplified and I felt like the people in the house were pressing in on me. Fortunately my wife noticed my discomfort and we left shortly afterward.

I had an even worse reaction earlier this year when we took our kids to an Easter egg hunt sponsored by the local fire department. The various hunts were divided into age groups and my wife assigned me to supervise our four year old while she went with Son of Atomic Spud. Due to poor weather, the event was held in a local school, so I found myself crammed into a room with literally hundreds of people and their children. Although I stayed in the most open parts of the room that I could find, by the time the hunt was over and we had streamed out of the building, my teeth were chattering and I was shaking. I admit that such an extreme reaction may be less introversion and more agoraphobia.

Yeah, it's kind of like this sometimes

At the same time, most introverts don't avoid social interaction altogether. Every couple weeks I play Warhammer 40,000 with several friends. Once a month three of us even watch an episode of Mystery Science Theater 3000 before the game. The difference between these activities and the Christmas party a few years back is that, first, there are usually only four or five of us in the room during these games. Once or twice we've had up to six people, which I think is my limit. Second, all of us have relatively similar personalities. Given that engineers are considered "archetypal" introverts, it's not surprising that our five core players (i.e., the "40K Cabal") are all engineers or scientists.

I have read of parents that are concerned about their right-brained (i.e., typically introverted) child's apparent difficulty with social interaction who are amazed at how animated and open he or she becomes when interacting with other introverted children. Personally, I have never had more than two or three good friends at any given time. In fact, as suggested by number 10, I think I could define various periods of my life based on who that one best friend was at the time. While I'm silent and closed around strangers (or even when I'm surrounded by too many people that I would call friends), the few close friends I've had can testify that I have plenty to say.


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