Sunday, June 28, 2020

Bending the Curve - Adding a Long Tail to Your Random Rolls

   The classic 2d6 and 3d6 die roll conventions provide a normal or Gaussian distribution, commonly referred to as a Bell Curve.  This is great when we are looking to simulate an evenly distributed range of scores, such as the classic six characteristics, but what about when we would like to randomly determine an NPC level?  That presents a use where the the bulk of the results should be towards the lower bound, while the upper bound is potentially indeterminate, in other words an asymmetrical curve.

     One of the classic asymmetrical curves is the Pareto Curve, which has been popularized as the 80-20 rule, where in management 80% of your problems are caused by 20% of your people; or 20% of your people do 80% of the work.  Helpdesk ticket closure times are another good example, where the bulk of the them are closed in a fairly short range of time, but others, especially those requiring equipment replacement, extend the overall time range far to the right.  Back when I was at GE Fleet Services, I would check the 3rd quantile, or the mark at which the technician had closed 75% of their tickets, rather than their average time to close.

     Looking at a 2d6 distribution we find, 97.22% probability that you roll less than a 12; 91.66% less than an 11;  83.53% less than a 10 and 72.42% less than a 9.  We can create a reasonable long tail distribution very simply.   Whenever an 11 or 12 is rolled, roll another d6 and add it to 10, producing a 2-16 distribution.


     3d6 distributions give me the opportunity to illustrate that you don't need to stay with the same size die when creating a long tail.   You have a 83.8% chance of rolling less than a 14 on 3d6.   So to create the long tail, I would roll another die whenever the 3d6 is 15 or more and add it to 14.   If I use a d6, it only gives a 3-20 distribution, not much of an increase over 3-18.  By using a d10 instead, I create a 3-24 distribution range, producing a significant tail at the high end.

     I used anydice to look at the probabilities.  Their calculator will allow you to see the percentages for any combination of dice.  

     I haven't come up with as simple a way to do a long left hand tail, but I don't have a use case either.  If I get one I'll probably just flip the scale at the bottom and read from right to left.

     As I said, the use case I identified is random NPC levels, I'd be interested in hearing of any other use cases you might come up with.

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