Dungeon Dice is a board game that Parker Brothers produced in the 1970s. Each player is a prisoner in a dungeon staging an escape, and needs to roll dice to tunnel their way out. What series of events caused you to be jailed in the first place is left to the players' imaginations.
Playing this the other night, my friend Catherine rolled all different symbols (the best combination you can roll, earning you three cards toward your freedom) three times in one game. I became curious about the probability of this and decided to figure it out here.
On any given turn, you roll six dice. To figure out the probability of rolling all different symbols, imagine you are rolling one die at a time. The first die can be anything (6/6 probability), the second die needs to be anything except for the first symbol that was rolled (5/6 probability), the third needs to be anything except the first two (4/6 probability), and so on until the last die, which has a 1/6 chance of being different from all the others.
To find the chance of all of these happening in a row, we multiply those probabilities. Here's a screenshot of a program called Jupyter (powered by the Python programming language) I used for this:
So the odds are 1.54% of doing this on any given die roll. But what about the odds of doing this three times in one game?
I believe the game lasted five rounds, so we want the probability of doing this at least three times in five rounds. A quick Google search turned up this explanation of how to calculate binomial distributions. It's simple:
It gets into Stats 400 series territory here, but fortunately the Python scipy library has some helper functions for this. This is the calculation of this happening exactly 3/5 times is:
Or 0.00356%. Adding in the probabilities of it happening 4 or 5 times only changes this to 0.00359%, or 1 in 27,850.
As a check of all this, it should be somewhat more likely than the chances of doing this three times out of three (whose probability is much easier to compute). Multiplying our 1.54% chance by itself three times yields 0.000367%, ~one tenth as likely as the chances of doing it 3 out of 5 times, so our result above seems sane.
Conclusion: Catherine is very good at Dungeon Dice.
