How do I calculate how many of an item I'll have in this inventory system?

Question

I beta tested a game last week with a pretty fun system. This system is called "Dés d'usure" (in French, because it will be a French game). I translate it into "Wear dice", but I'm unsure.

When you have an item, you don't know how many copy of this item you have. Instead you're given a dice vaguely representing your quantity of the item. After using the item you roll to see if you keep the same quantity, or if you go down a stage (1d10 → 1d8 → 1d6 → 1d4). If you go down a stage from 1d4, you have none left.

Let's give an example.

The GM says you have "1d10" potions. Now you use a potion, you apply the effect, then you roll 1d10. If the result is more than 3, you still have "1d10" potions. But if the result is between 1 and 3, you go down one stage and now only have "1d8" potions.

Now that you're at "1d8" potions, if you use another potion, you roll 1d8. If the result is more than 3, you still have "1d8" potions. But if the result is between 1 and 3, you go down a stage: you now have "1d6" potions.

The system goes until 1d4. If you roll between 1 and 3 with 1d4, you don't have any potions left in your bag.

---

So let's say I have "1d10" potions in my bag, how can I calculate the average of how many potion I will have?

Long story: I really want to re-use this system for a homemade RPG. But I'm questioning myself if I keep the "1-3" limit or if I will use a different limit like "1-2". But I wonder how this will modify the probabilities.

Answer 1

If you're in "Stage d10", you have a \$\frac{7}{10}\$ chance to stay there, and \$\frac{3}{10}\$ to go to "Stage d8". So the average number of potions in this stage is \$1 + \frac{7}{10} + \frac{7}{10}^2 + \frac{7}{10}^3 + \ldots = \frac{10}{3}\$. If you wonder where that number is coming from: it's from the well-known geometric series, which say that this sum is equal to \$\frac{1}{1-\frac{7}{10}}\$.

If you're in "Stage d8", you have a \$\frac{5}{8}\$ chance to stay there, and \$\frac{3}{8}\$ to go to "Stage d6". So the average number of potions in this stage is \$1 + \frac{5}{8} + \frac{5}{8}^2 + \frac{5}{8}^3 + \ldots = \frac{8}{3}\$.

If you're in "Stage d6", you have a \$\frac{1}{2}\$ chance to stay there, and \$\frac{1}{2}\$ to go to "Stage d4". So the average number of potions in the stage is \$1 + \frac{1}{2} + \frac{1}{2}^2 + \frac{1}{2}^3 + \ldots = 2 = \frac{6}{3}\$.

If you're in "Stage d4", you have a chance of \$\frac{1}{4}\$ to stay there. Therefore, the average number of potions in this stage is \$1 + \frac{1}{4} + \frac{1}{4}^2 + \frac{1}{4}^3 + \ldots = \frac{4}{3}\$.

If we sum this, we end up with \$\frac{10}{3} + \frac{8}{3} + \frac{6}{3} + \frac{4}{3} = 9\frac{1}{3}\$.

---

Another way to modify the probabilities (rather than switching to 1-2) is to start at another die, e.g. Stage d8 instead of Stage d10. In general, lower 'thresholds' (your proposed 2 is lower than 3) increase the average, and lower values for the starting die decrease the average number of items. To calculate this quickly, we have the following formula (thanks @RyanThompson for the hint):

The average number of potions in stage d\$N\$ if you need to beat \$m\$ is \$\frac{N}{m}\$.

In your original case, \$m=3\$.

Answer 2

As other answers have pointed out, the number of items in each stage is a geometric random variable. The total number of items across stages is therefore a sum of geometric random variables. Because each of the variables comes from a different geometric distribution, the distribution on the sum is a bit complicated, but we can make some progress by considering the distribution of each geometric random variable in isolation. Specifically, we can understand the mean and the variance of the sum if we understand the mean and variance of each random variable. Understanding the variance may be important if you want to avoid dice mechanics that are too noisy.

The distribution of items in individual stages

A geometric random variable is defined by the parameter \$p\$, which in this case is the probability of going down one stage. For stage \$i\$, we'll call this probability \$p_i = m \div N_i \$ (where \$m\$ is the number to beat and \$N_i\$ is the size of the die at stage \$i\$), and the (random) number of items in the stage \$X_i\$. The expected number of items in stage \$i\$ is: $$ \mathrm{E}(X_i) = \frac{1}{p_i}, $$ and the variance is: $$ \mathrm{Var}(X_i) = \frac{1 - p_i}{p_i^2}. $$

Intuitively, as the probability increases, the expected value decreases. Likewise, as the probability increases, the variance decreases.

The mean and variance of the total number of items

The expected value of the sum of \$n\$ random variables is the sum of the expected values of each random variable. Therefore, the expectation of the sum, \$S\$, is: $$ \mathrm{E}(S) = \sum_{i=1}^n \frac{1}{p_i}. $$ The variance of the sum of independent random variables (which these are) is the sum of the variances of each random variable: $$ \mathrm{Var}(S) = \sum_{i=1}^n \frac{1 - p_i}{p_i^2}. $$ The takehome message here is that if you decrease the probability of each stage (i.e., decrease \$m\$), the total number of items will become more noisy. This may or may not be desirable.

The probability of \$s\$ items

With some effort, we can calculate the probability that the total number of items across \$n\$ stages is equal to some particular value, \$s\$: $$ P(S = s \mid p_1, p_2, \ldots, p_n) = \sum_{i=1}^n p_i(1-p_i)^{s-1} \prod_{j=1, j \neq 1}^n \frac{p_j}{p_j - p_i} $$ for \$s \geq n\$. This is adapted from equation (1) in this paper. I plot this function (and the mean and variance) for your particular sequences \$(1\mathrm{d}10 \rightarrow 1\mathrm{d}8 \rightarrow 1\mathrm{d}6 \rightarrow 1\mathrm{d}4)\$ and for various values of \$m\$:

Answer 3

Just for fun (and to confirm Glorfindel's calculations), let's model this in AnyDice:

MAXROLLS: 40
set "maximum function depth" to MAXROLLS

function: iterate DIE:d {
  result: 1 + (DIE > 0) * [iterate DIE]
}

TOTAL: 0
loop SIDES over {4, 6, 8, 10} {
  TOTAL: [iterate 1dSIDES > 3] + TOTAL
  output [lowest of MAXROLLS and TOTAL] named "total items left at d[SIDES] stage"
}

The core of this code is the function [iterate DIE], which takes an arbitrary AnyDice die (i.e. a probability distribution over the integers) and returns the distribution of the number times (from 1 to MAXROLLS) one needs to roll the die until one rolls 0 (or less).

In AnyDice, a comparison involving dice, like 1d10 > 3, just return a custom die that rolls 1 with probability of the comparison being true, and 0 otherwise. So calling [iterate 1d10 > 3] directly gives us the distribution of the number of rolls until the player drops from the d10 stage to the d8 stage.

The rest of the code then just loops over the available stages (from the lowest up) and sums up the total number of rolls the player can expect to perform until dropping down to zero items left. Here's what the results look like:

(Since each individual stage is truncated to at most MAXROLLS rolls, I also explicitly truncate the total roll counts to the same limit. That makes the plot more readable, and makes any truncation bias easier to see by ensuring that we have at least as many rolls available for each individual stage as we have in total.)

And yes, we can see that the numerical averages calculated by AnyDice indeed match Glorfindel's math: each additional d\$N\$ stage gives you an average of \$N/3\$ more items before running out. The extra value AnyDice gives is also showing the distribution of the results around the mean, which has quite a long exponentially decaying tail.

(Of course, we could also plot these distributions without using tools like AnyDice, by noting that the number of rolls at each stage is independently geometrically distributed, and looking up the appropriate formulas for that distribution. But that would be way more work.)

Answer 4

Interesting question!

My answer is a little late, and comes to the same conclusions as the others, but I wanted to include it as it provides some exact values which are missing in the above.

I focused on the distribution of chance to obtain each number of uses for your die. Included some graphs for completeness.

This was calculated by creating infinite series where the coefficient of x^n corresponds to the probability of obtaining n uses of a potion. I expand the polynomial by terms.

e.g.:

• d4 = (3/4 * x) * sum(i=0..inf) (x * 1/4)^i
• d6 = (3/6 * x) * sum(i=0..inf) (x * 3/6)^i * d4
• d8 = (3/8 * x) * sum(i=0..inf) (x * 5/8)^i * d6
• d10 = (3/10 * x) * sum(i=0..inf)(x * 7/10)^i * d8

And calculate the coefficient using some multiplication (roughly equivalent to the below python):

def _coef(n, *args):
    # Sum all possible products such that the total exponent adds to n
    # This can be calculated recursively, e.g.:
    # f(2,  a,b,c) -> a^2 + ab + ac + b^2 + bc    + c^2
    #                 a(a+b+c)      + b(b+c)      + c(c)
    #                 a*f(1, a,b,c) + b*f(1, b,c) + c*f(1, c)
    if n <= 0:
        ret = 1
    elif n == 1:
        ret = sum(args)
    else:
        total = 0
        for i, a in enumerate(args):
            total += a * _coef(n - 1, *args[i:])
        ret = total
    return ret

def coef(n, *args):
    if n < len(args):
        return 0
    fail_factor = product((1 - a) for a in args)
    coef_factor = _coef(n - len(args), *args)
    return fail_factor * coef_factor

My apologies to mobile users.

For die d4

+--------+---------------+--------------+---------------------+--------------------+
|   Uses |   Probability |   Cumulative | Probability Exact   | Cumulative Exact   |
+========+===============+==============+=====================+====================+
|      0 |    0          |     0        | 0                   | 0                  |
+--------+---------------+--------------+---------------------+--------------------+
|      1 |    0.75       |     0.75     | 3/4                 | 3/4                |
+--------+---------------+--------------+---------------------+--------------------+
|      2 |    0.1875     |     0.9375   | 3/16                | 15/16              |
+--------+---------------+--------------+---------------------+--------------------+
|      3 |    0.046875   |     0.984375 | 3/64                | 63/64              |
+--------+---------------+--------------+---------------------+--------------------+
|      4 |    0.0117188  |     0.996094 | 3/256               | 255/256            |
+--------+---------------+--------------+---------------------+--------------------+
|      5 |    0.00292969 |     0.999023 | 3/1024              | 1023/1024          |
+--------+---------------+--------------+---------------------+--------------------+

For die d6

+--------+---------------+--------------+---------------------+--------------------+
|   Uses |   Probability |   Cumulative | Probability Exact   | Cumulative Exact   |
+========+===============+==============+=====================+====================+
|      0 |   0           |     0        | 0                   | 0                  |
+--------+---------------+--------------+---------------------+--------------------+
|      1 |   0           |     0        | 0                   | 0                  |
+--------+---------------+--------------+---------------------+--------------------+
|      2 |   0.375       |     0.375    | 3/8                 | 3/8                |
+--------+---------------+--------------+---------------------+--------------------+
|      3 |   0.28125     |     0.65625  | 9/32                | 21/32              |
+--------+---------------+--------------+---------------------+--------------------+
|      4 |   0.164062    |     0.820312 | 21/128              | 105/128            |
+--------+---------------+--------------+---------------------+--------------------+
|      5 |   0.0878906   |     0.908203 | 45/512              | 465/512            |
+--------+---------------+--------------+---------------------+--------------------+
|      6 |   0.0454102   |     0.953613 | 93/2048             | 1953/2048          |
+--------+---------------+--------------+---------------------+--------------------+
|      7 |   0.0230713   |     0.976685 | 189/8192            | 8001/8192          |
+--------+---------------+--------------+---------------------+--------------------+
|      8 |   0.0116272   |     0.988312 | 381/32768           | 32385/32768        |
+--------+---------------+--------------+---------------------+--------------------+
|      9 |   0.00583649  |     0.994148 | 765/131072          | 130305/131072      |
+--------+---------------+--------------+---------------------+--------------------+
|     10 |   0.00292397  |     0.997072 | 1533/524288         | 522753/524288      |
+--------+---------------+--------------+---------------------+--------------------+
|     11 |   0.00146341  |     0.998536 | 3069/2097152        | 2094081/2097152    |
+--------+---------------+--------------+---------------------+--------------------+
|     12 |   0.000732064 |     0.999268 | 6141/8388608        | 8382465/8388608    |
+--------+---------------+--------------+---------------------+--------------------+

For die d8

+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|   Uses |   Probability |   Cumulative | Probability Exact                 | Cumulative Exact                      |
+========+===============+==============+===================================+=======================================+
|      0 |   0           |     0        | 0                                 | 0                                     |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      1 |   0           |     0        | 0                                 | 0                                     |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      2 |   0           |     0        | 0                                 | 0                                     |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      3 |   0.140625    |     0.140625 | 9/64                              | 9/64                                  |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      4 |   0.193359    |     0.333984 | 99/512                            | 171/512                               |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      5 |   0.182373    |     0.516357 | 747/4096                          | 2115/4096                             |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      6 |   0.146942    |     0.6633   | 4815/32768                        | 21735/32768                           |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      7 |   0.108868    |     0.772167 | 28539/262144                      | 202419/262144                         |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      8 |   0.076694    |     0.848861 | 160839/2097152                    | 1780191/2097152                       |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|      9 |   0.052294    |     0.901155 | 877347/16777216                   | 15118875/16777216                     |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     10 |   0.0348724   |     0.936028 | 4680495/134217728                 | 125631495/134217728                   |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     11 |   0.0228917   |     0.958919 | 24579819/1073741824               | 1029631779/1073741824                 |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     12 |   0.0148561   |     0.973775 | 127613079/8589934592              | 8364667311/8589934592                 |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     13 |   0.0095596   |     0.983335 | 656930547/68719476736             | 67574269035/68719476736               |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     14 |   0.00611204  |     0.989447 | 3360131775/549755813888           | 543954284055/549755813888             |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     15 |   0.00388868  |     0.993336 | 17102611899/4398046511104         | 4368736884339/4398046511104           |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     16 |   0.00246476  |     0.995801 | 86720945319/35184372088832        | 35036616020031/35184372088832         |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     17 |   0.00155764  |     0.997358 | 438436417347/281474976710656      | 280731364577595/281474976710656       |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     18 |   0.000982107 |     0.99834  | 2211509144655/2251799813685248    | 2248062425765415/2251799813685248     |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     19 |   0.000618109 |     0.998958 | 11134854544779/18014398509481984  | 17995634260668099/18014398509481984   |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+
|     20 |   0.000388464 |     0.999347 | 55983509189559/144115188075855872 | 144021057594534351/144115188075855872 |
+--------+---------------+--------------+-----------------------------------+---------------------------------------+

For die d10

+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|   Uses |   Probability |   Cumulative | Probability Exact                                                                 | Cumulative Exact                                                                      |
+========+===============+==============+===================================================================================+=======================================================================================+
|      0 |   0           |    0         | 0                                                                                 | 0                                                                                     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      1 |   0           |    0         | 0                                                                                 | 0                                                                                     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      2 |   0           |    0         | 0                                                                                 | 0                                                                                     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      3 |   0           |    0         | 0                                                                                 | 0                                                                                     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      4 |   0.0421875   |    0.0421875 | 27/640                                                                            | 27/640                                                                                |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      5 |   0.0875391   |    0.129727  | 2241/25600                                                                        | 3321/25600                                                                            |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      6 |   0.115989    |    0.245716  | 118773/1024000                                                                    | 251613/1024000                                                                        |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      7 |   0.125275    |    0.370991  | 5131269/40960000                                                                  | 15195789/40960000                                                                     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      8 |   0.120353    |    0.491344  | 197186157/1638400000                                                              | 805017717/1638400000                                                                  |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|      9 |   0.107255    |    0.598599  | 7029078021/65536000000                                                            | 39229786701/65536000000                                                               |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     10 |   0.0907668   |    0.689366  | 237939825213/2621440000000                                                        | 1807131293253/2621440000000                                                           |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     11 |   0.0739985   |    0.763364  | 7759306121589/104857600000000                                                     | 80044557851709/104857600000000                                                        |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     12 |   0.0586665   |    0.822031  | 246065046795117/4194304000000000                                                  | 3447847360863477/4194304000000000                                                     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     13 |   0.0455234   |    0.867554  | 7637554195028901/167772160000000000                                               | 145551448629567981/167772160000000000                                                 |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     14 |   0.0347342   |    0.902288  | 233097529579949853/6710886400000000000                                            | 6055155474762669093/6710886400000000000                                               |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     15 |   0.0261476   |    0.928436  | 7018937631217111509/268435456000000000000                                         | 249225156621723875229/268435456000000000000                                           |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     16 |   0.0194699   |    0.947906  | 209056580748542012877/10737418240000000000000                                     | 10178062845617497022037/10737418240000000000000                                       |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     17 |   0.0143684   |    0.962274  | 6171165847820748626181/429496729600000000000000                                   | 413293679672520629507661/429496729600000000000000                                     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     18 |   0.0105251   |    0.972799  | 180820654310520268173693/17179869184000000000000000                               | 16712567841211345448480133/17179869184000000000000000                                 |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     19 |   0.00766224  |    0.980462  | 5265448029983050174879029/687194767360000000000000000                             | 673768161678436868114084349/687194767360000000000000000                               |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     20 |   0.005549    |    0.986011  | 152529676741737471547003437/27487790694400000000000000000                         | 27103256143879212196110377397/27487790694400000000000000000                           |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     21 |   0.00400084  |    0.990012  | 4398967032423950869575861861/1099511627776000000000000000000                      | 1088529212787592438713990957741/1099511627776000000000000000000                       |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     22 |   0.00287374  |    0.992885  | 126388634748253166004618272733/43980465111040000000000000000000                   | 43667557146251950714564256582373/43980465111040000000000000000000                     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     23 |   0.00205751  |    0.994943  | 3619603834210652189541665152149/1759218604441600000000000000000000                | 1750321889684288680772111928447069/1759218604441600000000000000000000                 |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     24 |   0.00146901  |    0.996412  | 103372621197087349842475462150797/70368744177664000000000000000000000             | 70116248208568634580726952600033557/70368744177664000000000000000000000               |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     25 |   0.00104632  |    0.997458  | 2945139485679173008972033887487941/2814749767106560000000000000000000000          | 2807595067828424556238050137888830221/2814749767106560000000000000000000000           |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     26 |   0.000743706 |    0.998202  | 83733822826925024585784972531302973/112589990684262400000000000000000000000       | 112387536535963907274107790488084511813/112589990684262400000000000000000000000       |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     27 |   0.000527654 |    0.998729  | 2376340268332105196766179822917498869/4503599627370496000000000000000000000000    | 4497877801706888396161077799346297971389/4503599627370496000000000000000000000000     |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+
|     28 |   0.000373775 |    0.999103  | 67333264212391058218558049842715358957/180143985094819840000000000000000000000000 | 179982445332487926904661670023694634214517/180143985094819840000000000000000000000000 |
+--------+---------------+--------------+-----------------------------------------------------------------------------------+---------------------------------------------------------------------------------------+