Here's one option: limit the number of chaos rolls and the number of planeswalk rolls to 1 per turn.
If you roll the planar die n
times, the probability of rolling at least one planeswalk symbol is 1-(5/6)^n
, and of course the probability is the same for chaos. So, my suggestion is this: determine the number of rolls (and tap for that much mana), then calculate that probability and round to the nearest 1/20. Then we can map those probabilities onto a D20.
For example, say I have enough mana to pay for 5 rolls. Then the probability of success is 0.598
, so we round to 0.60
. Then we can say that the "bottom" 12 numbers (1-12) correspond to planeswalk and the "top" 12 numbers (9-20) correspond to chaos. 9-12, of course, correspond to both. In those cases, flip a coin to determine which happens first.
Note that this severely inflates the probability that something happens. In the previous example, if I actually rolled the planar die 5 times, there would be about a 0.13
chance of nothing happening. However, it also removes the possibility of either planeswalk or chaos happening more than once.
For reference, here are the success rolls on a D20 corresponding to various numbers of planar die rolls:
- Just use a planar die
- planeswalk:
1-6
. chaos: 15-20
- planeswalk:
1-8
. chaos: 13-20
- planeswalk:
1-10
. chaos: 11-20
- planeswalk:
1-12
. chaos: 9-20
- planeswalk:
1-13
. chaos: 8-20
- planeswalk:
1-14
. chaos: 7-20
- planeswalk:
1-15
. chaos: 6-20
- planeswalk:
1-16
. chaos: 5-20
- planeswalk:
1-17
. chaos: 4-20
- planeswalk:
1-17
. chaos: 4-20
- planeswalk:
1-18
. chaos: 3-20
- planeswalk:
1-18
. chaos: 3-20
- planeswalk:
1-18
. chaos: 3-20
- planeswalk:
1-19
. chaos: 2-20
- planeswalk:
1-19
. chaos: 2-20
- planeswalk:
1-19
. chaos: 2-20
- planeswalk:
1-19
. chaos: 2-20
- planeswalk:
1-19
. chaos: 2-20
- planeswalk:
1-19
. chaos: 2-20
Above 20, the probability for each rounds to 100%, so just flip a coin to see which happens first.