Very Superstitious...

Dice rolls balance out in the end...

How many times have you heard this one you are on the losing end of a snakeyes?

It's not true.

This is what is known as The Gambler's Fallacy.

We all know that over repeated samples, dice rolls do tend toward the median (7 on 2D6).  This is because the mean is also the mode (or most frequently occurring number.)  The effect is simply more pronounced if we use a coin flip because there is no mode.

The distribution of possible outcomes of 2d6 shows why the mean DR tends toward 7.  Not only is 7 the modal outcome, but likelihoods decline the farther one gets from 7.  In all, there is slightly more than a 44% chance of rolling either a 6, 7 or 8 on any given roll.

However, just because likelihoods stabilize with more repetitions (assuming fair dice), does not change the fact that the outcome of each DR is independent.  That is, the same probabilities exist for each roll.  Dice do not keep track of what prior outcomes are and produce results to balance out previous rolls.

The cumulative average DR will move toward 7, but that is solely because it is the modal, or most frequently occurring, result.  This tendency is reinforced by the fact that toward the extremes results are not only less likely but self-cancelling.  (e.g. 2 + 12 = 14/2 = 7)

In ASL, all DR are not the same.  A subset of the DR are throwaway rolls (TH for acquisition, wind change, etc.) where you don't care about the outcome.  Another subset are relevant but non crucial.  A far more important subset is the absolutely crucial DR which determine the outcome of a scenarios.

Because the probability distribution, modal and mean rolls is precisely the same, we consider these subsets to be irrelevant in terms of the dice "evening out."  However, because the sample size is smaller (I put forward that five DR are really crucial to the outcome of a scenario), the chances that the outcome will differ from the expected mean are much higher.  Not just higher, but geometrically higher.

I did an experiment where I rolled dice (actually, I generated random numbers to create DR.)  To simulate the crucial DR, I rolled the dice seven times and then repeated it, measuring the mean and the mode after each seven times.  I repeated this for 20 different sets of rolls.

As predicted, the deviation from the mean and the mode go down as we roll the dice more times.  This demonstrates how a fewer number of crucial rolls are less likely to "even out" than a larger number of rolls.  In fact, we can't be 95% sure of outcomes until after 112 rolls.  Whatever the number of crucial rolls, its far less than that.

The mean (average) is a descriptive statistic and not in any way to be considered a prediction of probability.  So dice rolls do not "average out."

My point is not to justify dice whining.  It is rather to demonstrate the gambler's fallacy of expecting dice to "even out."  Also, it might be interesting to set up a system of identifying crucial DR while in your game and record and analyze them.  For example, if you and your opponent agree that a roll is a "big roll" before you make the DR, record the roll on a sheet of paper.  If it is a "big roll" after the DR, record it separately.  Sometimes you know a roll is going to be important before you roll it (CC), but sometimes you don't (88 malfing on an acquisition shot.)  It's important to keep these separate.

Anecdotal evidence, that is, when someone tells a story about dice evening out, reinforces this misconception.  Because we don't remember all the rolls, just the important ones.  To illustrate, I simulate 10 ASL games where a player rolls the dice 100 times.  The numbers, as expected, only differ by .2 over 100 rolls.  However, if we randomly assign a roll as "important" 10% of the time, the deviation of the important rolls jumps to .6--more than half a pip.  Cutting this to 5% made no difference in my sample.

	10% Important			5% Important
Game	All Rolls	Important Rolls		All Rolls	Important rolls
1	6.9	8.5		7.3	9.2
2	7.1	7.3		6.9	6.2
3	6.9	6.4		7.3	6.3
4	7.2	7.6		7	6.8
5	7.2	7.3		7.3	6.8
6	7	8.4		6.7	6.7
7	7	6.3		6.9	6.1
8	6.7	6.5		7.3	7.1
9	7.3	6.9		7.1	7
10	6.3	6.6		7.2	6.2
	0.2	0.6		0.2	0.6

Two things.  In Boulder, we sometimes played with a Mulligan.  It was, I think, a little rubber pig from Pigmania.  If you held the pig, you could turn it over to your opponent and roll again.  So instead of malfing that 88, you could play the mulligan.  It didn't help in every case (sometimes your opponent had the mulligan when you needed it and sometimes the re-roll didn't help you.  This somewhat diminishes the "important roll" problem.  It also really pisses off Tom Jazzbutis.

Last weekend, I had an opponent tell me, after offing 6 of my tanks with critical hits, that his "strategy worked perfectly."  Um, yeah.  Funny how that works.  As I said, this analysis isn't to justify dice whining, but emphasize that when losing--and when winning--it helps your sportsmanship to remember that dice do not balance out.  That's the gambler's fallacy.

Remember:  It's not how low you roll, it's when you roll low.