is a role playing system. My RPG group started playing an adventure in it last night. We had a lot of fun, and the game has an interesting dice mechanic which features the ominous sounding "exploding dice" (for me, this was part of the fun).
Each character has several attributes (Strength, Faith, etc... think ability scores in D&D). However, instead of a number, each ability is represented as a die size (up to d12) (which I guess is a number, but you see the difference). So you can have d6 in Strength, d10 in Faith, d4 in Smarts etc etc.
To resolve the outcome of an action, you roll two dice. The first is the attribute die for the appropriate attribute, and the second is the "wild" die which is always a d6. The higher value of the two rolls is kept. However, (and this is the interesting part) there is an "exploding die" rule. If you roll the maximum value for a particular die (4 on a d4, 6 on a d6, 8 on a d8 etc), that die "explodes" which means that you roll the die again, and add that result to previous result. If you roll another max on the die, the die "explodes" again and so on. If both dice explode, then you re-roll both, but add keep the corresponding totals from corresponding dice. In other words, you usually roll the dice together, but the result you get is equivalent to rolling one of the dice (with explosions) and getting a total, then rolling the other die (with explosions) and getting a second total, then keeping the larger of the two totals.
So, for example, if the character described above decides to try to bash in a door (a strength based action), the player would roll 2d6. Suppose he gets a 4 and a 6. The 6 explodes, so he re-rolls, getting another 6 which he then re-re-rolls, getting a 3. The first die gives him a 4, the second die gives him a 6+6+3=15. The higher result is 15. If the toughness of the door is less than 15, then the action is a success, if not, the character bounces off the door and crumbles into a heap of broken bones and spirits.
Another example: if the same player wants to convert a heathen (a faith based action) he'll roll a d6 and a d10. This time, he gets a 6 on the d6 and a 10 on the d10. He explodes the 6, getting 6, then explodes again getting a 3. Exploding the 10 yields a 1. The higher result is 15 (the d6 yields 6+6+3 = 15, the d10 yields 10 + 1 = 11). If the heathen is friendly and not particularly strong willed, this would probably represent a success. If the heathen happens to be Christopher Hitchens
, then the action would probably result in a failure. As a side note, if you roll snake-eyes (ie 1's on both dice), that represents a critical failure. In this case, that might mean that Hitchens converts you to atheism instead (cue Yakov Smirnoff
joke about conversions in Soviet Russia).
At any rate, being the math geek of the group, I want to analyze what the expected outcome of a particular roll will be. Expected value is a weighted average of all the possible outcomes (each outcome is weighted according to the probability of that outcome). Here are some warm-ups if you (my hypothetical reader) wants to play along:
1. What is the expected result of making a single (ie non-exploding) roll on a d4? d6? d8? d10? d12? (Generalize to dN)( Collapse )
2. What is the expected result of rolling a (non-exploding) d4 and a d6, keeping the maximum result? (Generalize to dM and dN)( Collapse )
3. What is the expected result of an exploding d4? d6? d8? d10? d12? Generalize to (dN).( Collapse )
4. The exploding dice mechanic raises an interesting question. On smaller dice, you expect to roll smaller numbers. However, you also expect to explode more often. Is there ever an advantage, in terms of expected result in rolling an exploding M sided die instead of an exploding N sided die where M < N?( Collapse )
5. What is the expected result when you roll two exploding d6's and keep the higher result? (Generalize to dN... is there a pair of integers, M<N such that you would prefer the M sided die to the N sided die?)( Collapse )
And now, the difficult problem (for which I have yet to find a solution):
6. What is the expected value of rolling a d4 and a d6 with explosions?