Usage Die / Usage Dots probability chart help

[simonefelli]

Hello,
I hope you can help me solve this dice problem with Icepool (maybe this has already been answered, but I could not find any info).

In many TTRPGs, a "Usage Die" is used to track some consumable items in a PC's inventory.
A Usage Die (Ud) of an item has a number associated with it, indicating its size. For example, oil (Ud6) has a Usage Die of d6.
When that item is used, its Usage Die is rolled. If the result is 1 or 2, then the Usage Die is downgraded to the next smallest die in the following chain:

d20 → d12 → d10 → d8 → d6 → d4

When you roll a 1 or 2 on a d4 (the lowest die in the chain), the item is expended, and the PC has no more of it left.

I would like to make a probability chart like this: given the Usage Die and a number of rolls (number of rolls >= remaining steps on the chain), what is the probability of depletion?
For example, arrows have a d10 Usage Die (Ud10); what is the probability that in 6 rolls (6 >= 4 remaining steps on the chain starting from d10) the arrows are gone?

If this is possible, I'd like to expand this to cover also "Usage Dots," which is more or less the same idea.
In this case, the dice chain is of the same dice type, but the number of dice can vary.
For example, a dice chain of 3 Usage Dots (d6) would be:

d6 → d6 → d6

When you roll a 1 or 2 on a d6, you mark off a dot; after 3 marks, the item is depleted.

Thanks for your time and this wonderful library; any help is very much appreciated.

[HighDiceRoller]

Glad to get your question. Here's a solution:

from icepool import d

die_sizes = [4, 6, 8, 10, 12, 20]

def usage_die(size, explode_depth=None):
    if explode_depth is None:
        explode_depth = size * 3
    index = die_sizes.index(size)
    if index == 0:
        return 1 + (d(size) > 2).explode(depth=explode_depth)
    else:
        return 1 + (d(size) > 2).explode(depth=explode_depth) + usage_die(die_sizes[index - 1])

for size in [4, 6, 8, 10, 12, 20]:
    output(usage_die(size), f'usage d{size}')

def usage_dot(size, t, explode_depth=None):
    if explode_depth is None:
        explode_depth = size * 3
    return t @ (1 + (d(size) > 2).explode(depth=explode_depth))
    
output(usage_dot(6, 3), 'usage dot 3d6')

limit(None, 50)

Here we use d(size) > 2 to determine whether each usage consumed the die (False) or if we get to keep using it (True). Then we use explode() to keep rolling until we consume the die. We add 1 since we still get to use the item on the roll that consumes it. From there, it's either recursively adding up the number of usages of shrinking dice, or adding up the usages from identical dice using the @ operator.

You can try this in your browser here.

[simonefelli]

Thank you for the quick answer! I'll dig into the code, ... I'm still learning to use Icepool. Just one curiosity: how do you get the 3 on "explode_depth = size * 3" (I guess it's for limiting the explode() function).

[HighDiceRoller]

That's right. The number of usages per die follows a geometric distribution, for which the mean result is the die size divided by 2 (plus one), and the standard deviation approximately the same. So I estimated that 6 times the mean is a good-enough limit for most cases.