Odds for dice pool with exploding dice

Question

I'm designing a tabletop game with dice pool mechanic for combat resolution. You roll some d6 and each die that rolled at least 3 counts as a SUCCESS:

• If you roll more 1s than 6s, it's a fumble
• One or two SUCCESSES is a light wound
• Three SUCCESSES in one roll is a severe wound
• Six SUCCESSES is a critical wound
• For each 6 you roll you get a SUCCESS. Also, add another d6 to the pool for each 6 you rolled. Count each of these die as a SUCCESS regardless of the number thrown. Each of these can also explode on 6.

I have all but last point covered in the below function (direct link to AnyDice function). But I'm struggling with the last part, the exploding dice mechanic.

That is critical to evaluating the system, as without exploding dice fumble chances increase, and are nearly 1/3 which is way too high. Exploding dice will limit the likelihood of fumbles. [edit: no, it won't]

Any idea how to write this function?

FUMBLE: -1
MISS: 0
WOUND: 1
SEVERE: 2
CRITICAL: 3

function: evaluate ROLL:s {
 SUCCESSES: ROLL >= 3
 if [count 1 in ROLL] > [count 6 in ROLL] { result: FUMBLE }
 if SUCCESSES = 0 { result: MISS }
 if SUCCESSES = {1..2} { result: WOUND }
 if SUCCESSES = {3..5} { result: SEVERE }
 if SUCCESSES >= 6 { result: CRITICAL }
}

loop DICE over {1..6} {
 output [evaluate DICE d6] named "[DICE]d6"
}

Answer 1

There's a built-in explode function, but you have to use it before doing your other processing. Specifically, use FOOdBAR to have BAR evaluated as a sequence and used to generate FOO "dice" with arbitrary side counts and values. Here, DICEd([explode d6]) does the job, turning an exploded d6 into the base for the actual roll. (explode doesn't separate dice that you give it — you have to do that, or you'll get a 5d6 roll that only explodes, once, on 30.)

You can't compare sequences to a threshold to get a list or sum of values against that threshold — you'll just get the sequence summed and checked against the threshold. Instead, use [count VALUES in SEQUENCE] for each of the valid values (3-6). And as it turns out, since every 6 triggers a new die and all those dice count, you can just double-count 6s once exploded. So really that should be [count {3..6, 6} in ROLL].

The results, as best I can manage, aren't pretty. Each explosion will either fumble harder (1), make no progress against fumbling (2-5), or break even and get a new chance to do the same thing (6), and the more dice you start with, the more chances for explosions. There is no provision for fumbles to peter out in explosions, as one would expect from a great success; the only question is whether the chain ends with a 1 or not.

Answer 2

I took a crack at it in Python (I know it's not anydice. Whatever.). Turns out, even if we exclude exploded ones from counting towards the fumble, fumbles are still WAY too common in any reasonably-sized die pool (my test pool of 8d6 still turned up fumbles around half the time). Here's my Python script that excludes exploded ones:

# This is exploding D6s.
# Basically, you start with a single roll with x dice.
# A result of 4 or higher is a success.
# A result of 3 or lesser is a failure.
# If a 6 is rolled, you roll another die. It will always be a success.
# If that die is a 6, you roll yet another, and so on. This is the "exploding" part.
# However, if more 1's are rolled than 6's, there's a FUMBLE.
# Exploded dice don't count towards 1's when rolled, but do count toward 6's.
# Let's... rationalize that.

import random

def explodeD6(x, xcount=0, explodes=0, explodecount=0, successes=0, ones=0): #x is the number of dice initially rolled.
  if xcount < x: #This means not all the dice in the initial pool have been rolled yet.
    roll = random.randint(1,6)
    if roll == 1:
      ones = ones + 1
    elif roll == 2 or roll == 3:
      successes = successes #Just demonstrating that nothing happens.
    elif roll == 4 or roll == 5:
      successes = successes + 1
    elif roll == 6:
      successes = successes + 1
      explodes = explodes + 1
    else: #Whoops! The randint wasn't between one and six. That's not right.
      print("Oops. Wrong die.")
    print(str(roll)) #Report.
    xcount = xcount + 1 #Indicate one more die has been rolled.
    explodeD6(x,xcount,explodes,explodecount,successes,ones) #And again.
  elif xcount == x: #This means all the initial dice have been rolled, but not necessarily the explosions.
    if explodecount < explodes: #This is an explosion roll. There are no failures, but it can explode again!.
      roll = random.randint(1,6)
      if roll == 1 or roll == 2 or roll == 3 or roll == 4 or roll == 5:
        explodes = explodes #Again, demonstrating nothing happens.
      elif roll == 6:
        explodes = explodes + 1
      else: #D'oh. randint still not working.
        print("Oops. Wrong die.")
      successes = successes + 1
      explodecount = explodecount + 1
      print(str(roll) + "(Explosion)") #Report.
      explodeD6(x,xcount,explodes,explodecount,successes,ones)
    elif explodecount == explodes: #All dice have been rolled and all manner of bonus dice shall have been rolled.
      totalRolls = x + explodes
      if ones > explodes: #Fumble.
        print("With %s ones, but only %s sixes, the result is a fumble, despite %s successes.") % (ones,explodes,successes)
        print("%s rolls total, from a starting pool of %sd6, due to %s explosions.") % (totalRolls,x,explodes)
      elif ones <= explodes:
        if successes > 0: #Success!
          print("There were %s ones, but %s sixes, allowing the %s successes to shine through.") % (ones,explodes,successes)
          print("%s rolls total, from a starting pool of %sd6, due to %s explosions.") % (totalRolls,x,explodes)
        elif successes == 0: #Miss.
          print("There were %s ones, but %s sixes, yet there were %s successes, resulting in a miss.") % (ones,explodes,successes)
          print("%s rolls total, from a starting pool of %sd6, due to %s explosions.") % (totalRolls,x,explodes)
        else: #Oops. Successes are negative.
          print("Something's wrong.")
    else:
      print("Something's wrong.")
  else:
    print("Something's wrong.")

#Holy shit. That's really long. Let's test it.

explodeD6(8)

Copy-paste it here to see it in action.

Answer 3

Looks like I was wrong and learned something. You can do this on AnyDice. :-) I tweaked that code in the first link in the comments to add in the logic to lower successes when 1s occur. It looks like with a 3d6 roll you would get a -1 or lower score ~8.5% of the time. 0 ~12%, 1 ~22% 2 ~22% and 3 or more ~35% of the time based on the results.

To clarify what this is doing: The functions at the bottom use 1s to indicate that they should allow for that thing (exploding dice, rerolls, subract fails) and 0s to indicate that they shouldn't. Threshold is the dice roll (or higher) that counts as a success. So if it is 3 then 3-6 are successes. I only added the bef function though you can easily tweak the functions to change threshold, if they explode or not, etc. The lion's share of the credit goes to @Magician who answered this question. So for your question you can just edit the last line bef 3 part to be bef [#d6 to roll initially].

function: roll ROLL:n threshold T {
  if ROLL >=T {result: 1}
  result: 0
}

function: rollexploding ROLL:n threshold T{
  if ROLL =6 {result: 1+[rollexploding 1d6 threshold T]}
  result: [roll ROLL threshold T]
}

function: rerollfailures ROLL:n threshold T{
  if ROLL < T {result: [roll 1d6 threshold T]}
  result: [roll ROLL threshold T]
}

function: subtractfailureexplode ROLL:n threshold T{
  if ROLL =6 {result: 1+[subtractfailureexplode 1d6 threshold T]}
  if ROLL =1 {result: -1+[roll ROLL threshold T]}
  result: [roll ROLL threshold T]
}

function: rerollthenexplode ROLL: n threshold T {
  if ROLL < T {result: [rollexploding 1d6 threshold T]}
  result: [rollexploding ROLL threshold T]
}

function: wrapper DICE:n threshold T explode E reroll R fail F{
  RES:0
  loop N over {1..DICE} {
    if E & F {RES:RES+[subtractfailureexplode 1d6 threshold T]}
      else {if E & R {RES:RES+[rerollthenexplode 1d6 threshold T]}
        else {if E {RES:RES+[rollexploding 1d6 threshold T]}
          else {if R {RES:RES+[rerollfailures 1d6 threshold T]}
            else {RES:RES+[roll 1d6 threshold T]}
        }
      }
    }
  }
  result:RES
}

function: b DICE:n{
  result:[wrapper DICE threshold 4 explode 0 reroll 0 fail 0]
}

function: be DICE:n{
  result:[wrapper DICE threshold 4 explode 1 reroll 0 fail 0]
}

function: gr DICE:n{
  result:[wrapper DICE threshold 3 explode 0 reroll 1 fail 0]
}

function: bef DICE:n{
  result:[wrapper DICE threshold 3 explode 1 reroll 0 fail 1]
}

output [bef 3]