Most Excellent Adventure is a home brew roleplaying game system based on the Bill & Ted Films, plays gnarly air guitar riff.
In this game system, when you draw from your dice pool you need to connect the results as a phone number on your phone pad:
Here one person has rolled 4, 2, 8, 3, 8, and 2, dialling 3-2-2-4-8-8. The other 7, 4, 6, 3, 9, and 4, only dialling 4-4-7 or 3-6-9. And the longest number wins.
How do I work out my chance for success (or chance of a certain phone number length) from this system?
I've tried enumerating the chance of getting a two digit number depending on which number you rolled first (each of the first ones is $1/10$) and I get:
But then, how do I follow each different 'route' of probabilities? I could imagine drawing a probability tree with ten branches and up to 12 levels, but that seems excessive. I could draw up a table (with blanks for non-telephonable combinations), but that would get hard to follow after the first table or so).
I've tried to consider combinatrics, but I've gotten myself confused over nPr and nCr notation and not getting the right numbers in there. Is there an easier way to calculate the probabilities?