Of Dice and Men

A recent discussion about standardizing dice at tournaments has prompted me to post some thoughts about dice, random numbers and the intrinsic biases that emerge.  I am going to start with a single basic premise:

Game design decisions are based on an expectation of a uniform probability distribution for dice rolls.

For those whom math is not a strong suit, I will explain what I mean by this.

Its All About the Dice

A uniform distribution in probability means there is an equal probability for any of the potential result.  While I am being a bit careless with my terminology (this is a discrete probability since the potential results are integers between 1 and 6 inclusive) the general idea is that the probability to roll a 1 should be the same as the probability to roll a 6.

 

Where am I going with this?  After the Da Boyz GT last year I had some horrendous luck in a few of my games.  In general I had a feeling that my dice rolls were very terrible for a while, but I started doing some research after the GT.  I was using the standard GW dice and a cube of Chessex Dice in my games.  I found some interesting research about both these dice at DakkaDakka - That's How I Roll 

 

I do not have access to the entire data set - but I will summarize the results here to say that over 144,000 rolls of 144 different dice the average number of 1's rolled for the data set was (removing statistical anomalies, which I am not sure what this means but it decreased the number) 29%.

 

This was astonishing to me, there should be 16% 1's for these dice.  They were rolling Chessex dice and GW dice - the same dice I was using.  From a statistical standpoint I wanted to see if these results were statistically significant - we all know that you can get lucky or unlucky, this test examines the probability that this result is actually different.

 

For example, if the dice are rolled 20 times - then some of the time I might roll 10 6's and it is perfectly reasonable.  However, as the number of rolls increases - the distribution should approach the theoretical value.
 

Random Numbers

To test for a discrete distribution a Chi-Squared test is used.  In this case, the parameters (for those math folks, others can skip this) are the observed frequency and expected frequency.  I am goin forward with the assumption there are two results - roll a 1 and everything else.  This is influenced by the available data - which is only the number of 1's rolled

 

Observed Frequency - [41760 102240]

Expected Frequency - [1/6 5/6] x 144000

 

Chi-Square: 2.27 x 10^9

 

This is a huge number and it means that the chance for 29% 1's over 144,000 rolls is virtually 0.  It must be then, that these dice do not roll each result with equal probability. (edit: there is more to this with respect to degrees of freedom but given the size of the Chi-square it does not matter)

 

I was shocked by this result, it threw off all the calculations and figuring I usually did.  It also really hurt daemons who (for example) were destroyed on a mishap 30% of the time rather than 16% of the time.  The same goes for Soul Grinders deep striking into terrain.

 

I resolved to get better dice - fair dice actually and purchased twenty Casino Dice for $40 from e-bay.  They are still available and I highly recommend them - there is nothing more amazing than rolling average!  Just search ebay for 'casino dice stick'.

 

 Why It Matters

 One proposed solution to this problem (and the problem of loaded dice) was to give each person the same set of dice to roll.  The premise is that if everyone has the same bad luck, then it must be fair.

 

The problem with this is that point costs are balanced around fair dice.  Poor rolling disproportionally effets different armies.  For example, a Terminator generally expects to fail 16% of saves.  If they actually fail 29% then they will fail 81% more saves  In contrast - a 5+ save (assume the other 5 results are equally as likely) takes a hit of 14% in this case.  The terminators are hugely more effected.

 

Throw in random charge ranges with this result, and charge distance shrink dramatically.  The odds to roll snake eyes goes from 1 in 36 to 1 in 9.  This is great for leadership tests but leaves assault elements in the open.  At the same time, BS 5 is undervalued (only a 70% hit rate) whereas large quantities of low quality firepower is less influence.  Generally speaking, MEQ armies and elite armies suffer more with dice biased towards rolling 1's.

 

What does this mean?  Tournaments need to think carefully before providing dice.  They need to provide high quality dice - or require a certainn type of dice if they are going to do so.

 

I will continue to use my casino dice, and am willing to let opponents use them as well - once.  After that they can rent them from me (these dice wear out) or get their own.

 

The real question for me is in a game so dependent on random numbers where we drop $40 on a Rhino Chassis - why would anyone be unwilling to purchase some dice with quality that matches the quality of models, terrain and display boards. 

 

For me, when those inevitable losses come, it is less about how bad the dice were and more about how bad a general I happened to be.