Sorry for the long post but it’s chart/math heavy. This is in relation to the Exploding Dice posting I found awhile back.
To sum up, use xd6 to do skill checks. Â If you roll a six add another d6 to the total. Â Keep doing this till you stop rolling six’s. Â Base is 3d6 for untrained, 4d6 for trained, 5d6 for trained + skill focus. This makes skill focus a better choice actually in my opinion than it was prior. It adds considerably more than the +10% that it adds under the existing rules.
Rather than compute a formula, that kind of statistics is beyond me, I just computed 100,000,000 skill check rolls using this system with a base of 3d6 through 6d6 (maybe the DM gave a bonus d6 for pastry tribute or something). The average roll for each is 12.6, 16.8, 20.99, 25.20. The highest rolls for each set run from around 60 to around 80. Even with such a large sampling the high rolls varied by several points per run as long runs of sixs are erratic if you roll 300-500 million of them.
I just knocked up a quick javascript to do it, it’s simple enough it worked on the second time. Any programmers out there know what I mean by that.
I computed the odds of DC’s between 5 and 25 inclusive for the untrained, trained, trained+skill focused:
Untrained aka 3d6:
DC 5 success rate 98%
DC 8 success rate 83%
DC 10 success rate 66%
DC 12 success rate 49%
DC 15 success rate 30%
DC 18 success rate 16%
DC 20 success rate 11%
DC 22 success rate 6%
DC 25 success rate 3%
Trained aka 4d6
DC 5 success rate 99%
DC 8 success rate 97%
DC 10 success rate 90%
DC 12 success rate 78%
DC 15 success rate 57%
DC 18 success rate 38%
DC 20 success rate 28%
DC 22 success rate 17%
DC 25 success rate 12%
Trained + Skill Focus aka 5d6
DC 5 success rate 100%
DC 8 success rate 99%
DC 10 success rate 98%
DC 12 success rate 94%
DC 15 success rate 81%
DC 18 success rate 63%
DC 20 success rate 52%
DC 22 success rate 36%
DC 25 success rate 27%
Perhaps a simple rule of thumb would be DC 10 if you want the majority of the time for the skill to succeed. Add 5 if you want to make it primarily trained characters to succeed. Add another 5 if you want really want it made only by supremely trained characters.
The above though does NOT count any attribute bonuses.
If we assume an average of +2 to any skill check the numbers for the untrained change as shown below.
3d6 + 2 from attribute modifier
DC 5 success rate 100%
DC 8 success rate 95%
DC 10 success rate 83%
DC 12 success rate 66%
DC 15 success rate 42%
DC 18 success rate 25%
DC 20 success rate 17%
DC 22 success rate 9%
DC 25 success rate 6%
If it’s a skill that’s a primary stat for the character and they get a +4 then you have the below and a DC 10 is almost guaranteed for even the untrained character.
3d6+4 from stat mod
DC 5 success rate 100%
DC 8 success rate 99%
DC 10 success rate 95%
DC 12 success rate 83%
DC 15 success rate 57%
DC 18 success rate 35%
DC 20 success rate 25%
DC 22 success rate 13%
DC 25 success rate 9%
5d6+4 from stat mod
DC 5 success rate 100%
DC 8 success rate 100%
DC 10 success rate 99%
DC 12 success rate 99%
DC 15 success rate 96%
DC 18 success rate 86%
DC 20 success rate 76%
DC 22 success rate 58%
DC 25 success rate 46%
So as a more possibly accurate formula this might work out:
DC 10
Add 0 to 4 for stat bonuses.
Add 2 per tier over heroic.
Add 5 if you want trained characters to shine.
Add 5 if you want suprememly trained characters to shine.
So we end up with DC’s between 10 and 28 (let’s say 25 since I only computed dc 25). Again we toss out the 1/2 skill level since as a rough rule DC’s go up by 1/2 level so again it’s a wash and makes no difference.
At the low end even an untrained without a stat bonus is going to succeed 2/3rds of the time and at the high end the same character still has a 3% of success.
And with training, excess training for some and stat bonuses a character is going to make even that hard check at DC25 almost half the time.
Anyway, I like the rule, going to house rule it in definitely because I disliked the ‘sorry, you’re one point short of being able to possible do this.’ factor of the existing skill chance. It’s always possible someone could get lucky or have a brain blast or ‘play above their abilities’. We all know the feeling of ‘being in the zone’ sometimes and doing things we couldn’t ever do again. I once one night while trying to impress a girl back in my misspent youth, sank 108 balls straight on a pool table. That’s a real pool table, not wii pool. 
A feat I’ve never even remotely come close to doing before or since. To put it in these terms my dice exploded and I rolled six after six. 
Under the standard DnD skill system I could never have done that.
