I've done some more thinking on this topic...
We were discussing an statistics problem here at work, and someone said "Why don't you ask that idiot on the internet" (Holcombs Arm). This, led to a discussion of dice with a true value of 3.5, and an investigation into the properties of such Holcombian dice, from which we determined this...
Prove, using Holcombian statistics, that half the time you don't roll a die, you get a value of 1.
Proof: a 6-sided Holcombian die has a Holcombian true value of 3.5. It is trivial to see that a Holcombian die with any other number of N sides will have a Holcombian "true" value of (N+1)/2 (ex: a 7-sided die has a Holcombian true value of 4, a 36 sided die has a Holcombian true value of 18.5, etc.)
From this, we can determine that the Holcombian true value of a Holcombian die with zero sides is actually one-half. However, if you try to "measure" the value of the zero-sided die, you will naturally get a result of zero. This represents, of course, an error of -0.5 in your measurement, which therefore means if you measure a zero-sided die a second time, the measurement will regress toward the mean, and you will measure a value of 1.
And since rolling a zero-sided die is mathematically equivalent to not rolling an N-sided die, it follows that, because error causes regression toward the mean, not rolling a die gives a value of 1 half of the time.
I'm sure there's many, many other interesting properties of Holcombian dice that follow from this...such as: a sixth of the time you roll a six-sided die, you get a one, which is equivalent to not rolling the die at all...therefore, rolling a die will regress to not rolling the die because of error...