On Expected Uses of Usage Dice

    I do have a page on useful RPG mechanics, ideas, and concepts on this blog and one of the concepts in there that of Dice Chain and Usage Dice. The central idea is this: Roll a die and if you roll at or below a specific number you drop down the chain. It gives a nice element of random chance to how many uses of something you get and makes tracking items like Arrows or Chalk very easy.

    However, it can be very unclear how many uses you would expect to get when you buy a Ud6 or Ud8 item. I have just gotten done doing some math, specifically using Markov Chains, and I believe my math holds up. Thus I have been able to produce this very handy chart:

Usage Die	Expected Uses
Ud4	2
Ud6	5
Ud8	9
Ud10	14
Ud12	20

    Now, that chart is certainly handy and can easily be recreated by noticing that each die adds 1/2 of its size to the number of expected uses. However, this default chart assumes a die decrements on a 1 or a 2. If you want more uses, you could just decrement dice on a 1. That, gives the following table:

Usage Die	Expected Uses
Ud4	4
Ud6	10
Ud8	18
Ud10	28
Ud12	40

    It is easy to notice that this table is exactly double the previous table, which makes sense since we halved the rate at which dice are decremented. If you just want to use this table on its own, you can easily recreate it by recognizing that each die adds its size to the expected number of uses.

    I don't have a lot of analysis to add to this, but I thought just knowing this would be helpful for people trying to balance Usage Dice in their own games. And, of course, I would like to know if you think my math is wrong.