I have a game system and need to work out some probabilities. Two dice are rolled, not necessarily with the same number of sides. Consider a d4 and a d10 for our example. If a die rolls its maximum it 'explodes'. That is, it rolls again and adds this to the total. I can do this indefinitely. So, for example, the d4 might get a total of 15 by rolling 4, 4, 4, 3. A d10 could do so with rolls of 10, 5. If the total of both rolls is 1, this is a Bad Thing, and the odds of this happening are fairly clear (in our example, 2.5% of the time, 25% for the d4 x 10% for the d10).
If it's not 1 on both rolls, then the higher roll is taken. So if the d4 rolls a 3 and the d10 rolls a 1, the result is 3. If the d4 rolls a 12 and the d10 rolls 11, it's 12. But at this point I don't know what the odds of rolling exactly 2 are, or exactly 3, or 4, or 12, or 22. I don't know how to figure that out. The probabilities of at least X would then be a simple matter of the 100-(Probability of X-1). But how do I figure out what the odds of getting exactly each of those numbers is?
Running a small program without the exploding dice (which is easier), I find that the values are as follows:
1 - 2.5%
2 - 7.5%
3 - 12.5%
5 - 17.5%
6 - 10%
7 - 10%
8 - 10%
9 - 10%
10 - 10%
I can't figure out how to get those numbers, either (without running through all possibilities, which I can't do with the exploding dice).
If it's not 1 on both rolls, then the higher roll is taken. So if the d4 rolls a 3 and the d10 rolls a 1, the result is 3. If the d4 rolls a 12 and the d10 rolls 11, it's 12. But at this point I don't know what the odds of rolling exactly 2 are, or exactly 3, or 4, or 12, or 22. I don't know how to figure that out. The probabilities of at least X would then be a simple matter of the 100-(Probability of X-1). But how do I figure out what the odds of getting exactly each of those numbers is?
Running a small program without the exploding dice (which is easier), I find that the values are as follows:
1 - 2.5%
2 - 7.5%
3 - 12.5%
5 - 17.5%
6 - 10%
7 - 10%
8 - 10%
9 - 10%
10 - 10%
I can't figure out how to get those numbers, either (without running through all possibilities, which I can't do with the exploding dice).