# Long Term Behavior of Dice

**11/19/05**

It is fairly well known that with "grocery store" dice, where material is scooped out of a side to form the divots to represent the number on the face, that in the long run the appearance of outcomes is biased. That is, the theoretical P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6 doesn't hold true because more material is scooped out of the sides with the larger outcomes, and one observes

My question is, with dice, still the "grocery store" variety, that have outcomes painted on, is it known that

in the long run?

If so, if (*) is observed at m rolls on average, at what number n, does (**) occur on average?

I think that with the cheap divoted dice one is able to observe the bias over their lifetime of rolling dice and watching others roll dice, even at an intuitive level if they don't keep detailed records. With painted dice I'm not sure, because the effect of weight on the bias due to paint added is probably much less than the effect of weight on the bias due to material being scooped out, and therefore I conjecture that n is **much** larger than m.

Of course good dice (more precision crafted, more durable materials, weighted paints, etc.) generally overcome the bias issues, but I was just thinking about the dice that most people use for casual everyday gaming.

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