Fair Dice

I came upon this really intriguing question here in the 11th Company forums.  The gist of the question is that if he "knows" his dice have different rolling percentages than average, is it fair to use them?  And when does it cross the line into cheating?  A very interesting question, indeed.  However, with the power of STATISTICS (fanfare), I believe the answer can become quite clear.

When you look at the data, it appears to be fairly irregular.  You should expect around 16-17% for each face of the die, and with one of his sets, he gets a percentage of 22%!  I won't get too much into the meat and potatoes of the statistics here, but I took the original poster's data and ran it through a Chi-square goodness of fit test (By the way, for all you haters, this sample size is plenty big).  Assuming the dice are fair to begin with (a standard assumption with any hypothesis test), there is a 12.13% chance that you would get similar results or a more extreme result.  This means that I have (roughly....) a 10% chance of any face showing up 22% of the time or more!  It's only for very large sample sizes (read: infinite sample sizes), where you expect to have exactly 16.67% proportions for each face of the die.

Just for fun, I did another goodness of fit test with the OP's second set of dice.  Instead of asking "what's the probability that I get a more extreme result" like I did last time, I asked "what's the probability of getting a more consistent result."  Our assumption that the dice are fair remain the same.  Guess what?  The probability that we get this level of a result or a more consistent result (that is, proportions closer to 16.67%) is only 8.78%.  It's very very rare to take a finite sample size and get perfect proportions.

To illustrate this point, I'd like to bring up a childhood anecdote.  On my TI-83 plus from high school, I got an application called "probability simulator."  When I was bored in class, I would call up this application and simulations.  Simulations with thousands of trials.  Despite this, I never got a perfectly uniform bar graph.  Why?  Because probabilities address only long-term proportions.  And long term does not mean a thousand trials.  It doesn't even mean a million trials.  Long term means an infinite number of trails.

Click to embiggen

That being said, let's talk about the cheating aspect of the question.  At what point would I consider this cheating?  My answer if very simple.  You're cheating if you brought unfair dice, knowingly or not.  If you make a decision based on an series of test rolls, you're probably just fooling yourself; there's nothing spectacularly unusual about the first set of dice.  And let's say that you engineered your dice so that sixes were rolled 20% of the time instead of 16.67%.  On average (that is, in the long-term), that means that you get an extra four sixes out of every hundred rolls of the die.  I don't believe that's enough of a chance to ruin a game.

Can you test for this?  Of course not!  That would be silly!  Nearly as silly as this xkcd.  (the "p" in the comic is the probability that you saw an increased chance of acne assuming that jelly beans are not correlated with acne.  .05 means 5%.   There.  Now you can read it and understand!).

Of course, the joke here is that they tested 20 different jelly bean colors and one of those tests was significant.  1/20 is 5%.  It's bound to happen that, if you do an experiment enough, you get the significant results.  That's the nature of the game.  And this would directly apply to a tournament testing dice.  If I have 64 people in a tournament and I'm testing at a 5% level, I would, on average, throw out 3 of them for having unfair dice, even if they had fair dice.


There's one last topic I'd like to address that was brought up in the thread.  That's the idea of "precision dice."  Why do people sell "precisions dice" if they're just as good as normal, cheapo dice?  Well, probably because people want to buy into it.  It's been shown time after time that tap water is at least as safe as bottled water (and/or that bottled water comes from a city's municipal water supply....in layman's terms...tap water), yet the bottled water industry is wildly successful.  It's all about how you sell your product, and Penn and Teller do a marvelous job of showing it.

Gay Rights and General Thoughts

I stumbled across the upcoming proposal to the North Carolina constitution today, which prompted an extended chain of videos and blogs, ending in some of my favorite anti-atheist videos and a search as to why the Christians are against gay marriage (from a legal and secular standpoint).

Peanut Butter depends on the lack of evolution!

I found a story of a man named David Parker who's son was exposed to the "gay agenda" in Kindergarten, where teachers were talking about families with two dads, same-sex couples, and so on.  David decided he didn't want his son exposed to this "radical" thinking, marched into the school to get some change happening, didn't get it, and was subsequently arrested for trespassing when he refused to leave.

Well then I'll just live in this tree school until I get what I want!

At first, the school's actions surprised me.  Normally we try to appease parents, and it's not unheard of to have parents request and grant requests for their children to "opt out" of evolution, much like David wanted to opt his child out of the gay curriculum (no pun).  The reason why David's request wasn't granted was because this is a basic civil rights issue.  The Catholic Church (at least in my experience) seeks to dehumanize those with different sexual identities.  I can't count the number of sermons I sat through with the priest professing the sins of being gay, the eternal hellfire which awaited them, and how gay people embodied the spirit of Lucifer himself, only to take everything back by saying, "If you are gay, we can save you!"

It got me to wondering what schools were like around the time of Brown v. Board of Education (desegregation of schools).  Were there 1950s versions of David who marched into their children's schools, demanding that interracial marriage (GASP) not be exposed to their poor innocent children?  Were there similar lawsuits and appeals filed to try to get children "opted out" of social change?

I just have one more thing to get off my chest, and I know it's a desperate plea and that it won't change anybody's mind, but here it goes...

The United States is a secular government.  Which is to say, it's not religious in nature.  Therefore, when you say things like "God said so" in support of a legal document, YOU'RE DOING IT WRONG.  Religious laws should be used to govern your personal life, not to govern the people of this country.  I have absolutely no problem with the Catholics saying that they don't want to marry gay couples.  Fine.  But when you use your religion as reasoning behind a legal action, it is against the spirit of the constitution of these United States.