Post/Author/DateTime	Post
flyingmoose 03-04-04, 10:28 PM	There's a rule in the EHB where if you roll a 20, you roll again and add 20. It's called open-ended rolling. The simple way to do this with 3d6 is if you roll an 18, roll again and add 18, but 18s on 3d6 a [darn] sight rarer than 20s on 1d20. Well other games do open-ended rolling with dice pools. This'd just be a fixed pool. 3d6, reroll any 6s and add 6. I have no idea how to calculate the odds on this (or I do, but the method I can think up would take too long). So I wrote a C++ program to roll a [poop]load of dice and come up with sort of a curve. (20,000 trials; rolls above 35: 51; each | represents 100 rolls, each . represents around 50) 32 31 30 . 29 . 28 . 27 | 26 |. 25 |. 24 || 23 ||. 22 ||| 21 ||| 20 ||||. 19 |||||. 18 |||||| 17 |||||||| 16 ||||||||. 15 ||||||||||. 14 ||||||||||| 13 ||||||||||||. 12 |||||||||||||. 11 |||||||||||||||| 10 |||||||||||||||||| 09 |||||||||||||||||| 08 |||||||||||||||| 07 |||||||||||||| 06 ||||||||| 05 ||||| 04 ||| 03 | Anybody interested?
Pystian 03-05-04, 08:24 AM	I thought it was cool.
MoonHawk 03-05-04, 01:57 PM	The curve looks interesting. I'd be even more interested to see actual percentages next to each bar, so I can see what points cross the 5% d20 mark, and what the maximum is and such. My only concern is that this might favor high rolls a bit much, but I'm not sure, that's why I want to see the numbers. Also, if you could run it again and come up with the percentages for non-exploding d6 so it's easy to compare to the standard 3d6 method, that would be a lot more useful.
MoonHawk 03-05-04, 02:01 PM	Wait a minute, I just realized that the curve peaks between 9 and 10. Even if the dice weren't exploding, you should still average 10.5, with exploding dice it should be even higher. Run more trials, if that average doesn't change, check your code. edit: You should find your peak at 12.6
SirisC 03-06-04, 12:04 AM	i dont use open ended but with 4d6 you dont drop any dice if you roll four of a kind. so you can get a 20 by rolling 4 5s and a 24 by rolling 4 6s (each has a 1 in 1296 chance of occuring) but i really like the open ended roll concept, but would only allow it for 4 of a kind not only would you add all together but you would roll all four again and add your previous total. (excluding rolling 4 1s)
Zubon 03-06-04, 03:14 PM	Originally posted by MoonHawk Wait a minute, I just realized that the curve peaks between 9 and 10. Even if the dice weren't exploding, you should still average 10.5, with exploding dice it should be even higher. Run more trials, if that average doesn't change, check your code. edit: You should find your peak at 12.6 Without running the numbers, it looks about right to me. The mean/median of that data probably is around 12.6, or nearabouts. The upper tail takes a long while to disappear, so that brings the average up from where the curve peaks. Some of our numbers that were going to be around 10.5 are shifted higher. Then again, I'm not running at full speed just now. I think we should see a mean around 12.6, not a peak. I could be wrong.
flyingmoose 03-07-04, 03:30 PM	Each one of those bars represents 0.5% of the trials. So 10 bars is 5%. How are you calculating your average? I'm not saying you're wrong, I'm not a particularly good programmer. I'd just like to see your calculations. Realize 6 and 7 have exactly the same probability (because you can't roll less than 1). I had it calulate a percent rather than spit out a string of bars, this is what I got. Also, this one was 30,000 trials, rather than 20,000. 3 0.466667% 4 1.33% 5 2.66333% 6 4.72667% 7 7.1% 8 8.22667% 9 9.05667% 10 9.01667% 11 8.01667% 12 6.81% 13 6.21667% 14 5.59333% 15 4.97667% 16 4.38667% 17 3.88667% 18 3.18667% 19 2.68% 20 2.33333% 21 1.59667% 22 1.5% 23 1.39333% 24 0.966667% 25 0.75% 26 0.696667% 27 0.493333% 28 0.366667% 29 0.393333% 30 0.24% 31 0.203333% 32 0.143333% 33 0.1% 34 0.11% 35 0.11% 36 0.0633333% 37 0.04% 38 0.03% 39 0.04% Rolls above 40: 22 (0.0733333%) (...is it just me or does the curve kinda look like a boob?)
flyingmoose 03-07-04, 03:31 PM	Would this affect threat ranges? Would you use the ones in UA or leave them unchanged or what?
MoonHawk 03-07-04, 04:20 PM	The way I was calculating my average was: The mean value of a d6 is equal to the probability of each value * the value itself, so: mean = 1/6 * 1 + 1/6 * 2 + 1/6 * 3 + 1/6 * 4 + 1/6 * 5 + 1/6 * 6 If the die is "exploding", then the last 6 is replaced with the value (6+mean). Factoring out the 1/6 term, you get: mean = 1/6 * (1+2+3+4+5+6+mean) 6 * mean = 21 + mean 5 * mean = 21 mean = 4.2 So your mean value should be 12.6 As for a mode value (i.e. the value that occurs the most) I don't have an exact calculation offhand, but consider this: The mode for a standard 3d6 is 10 and 11, both occur equally often, and more often than any other number. I'd be happy to try to calculate this, but it's been a while since my last probability course. But with exploding dice, it has to be higher than with non-exploding. I'll see if I can calculate the probability of each result explicitly, but like I said, it's been a while.
MoonHawk 03-07-04, 05:12 PM	Okay, here's what I get, using numerical methods: result - probability (%) 3 - 0.46 4 - 1.39 5 - 2.87 6 - 4.63 7 - 5.09 8 - 8.79 9 - 9.03 10 - 9.03 11 - 8.33 12 - 6.94 13 - 6.25 . . . I could go on, but in short, my numbers are similar to yours. The mean value of an exploding d6 is 4.2, but since a roll of 6 is not possible (since a six is always rerolled), certain combinations of multiple dice that are very probable with non exploding dice become slightly biased against. For example, we all know the probability of a 7 on 2d6 is 1/6, but with exploding dice the probability drops to 1/9, as some of that probability is shifted upward toward higher numbers. Summary for this experiment is: Mean: 12.6 (exactly) Median: irrelevant to this type of random process Mode: 9 and 10, each occuring with probability 13/144 (exactly) or about 9.03% of the time. I really like the way the probability ends up being distributed. Good work, and if you don't mind waiting for all that rolling it should give a very realistic dragon slaying experience. :uh-huh: edit: table formatting
Tam 03-07-04, 08:57 PM	at first glance (I may be wrong): Calling x the average on 1 dice: x = 3,5 +1/6*x solving: x = (3,5*6)/5 = 4,2 With three dices, that would put the average at 12,6.
full_plate 03-08-04, 12:07 AM	Hi I like statistics and all but what is the real advantage to using exploding dice instead of just 3d6? Sure there is the remote possibility of rolling like 257 and having a limbless commoner throw a dagger from his mouth and kill an ancient red wyrm dead in his tracks, but besides that why? It is still going to give you a normal curve while only slowing down combat with more dice rolling. And correct me if I am dum, but wouldn't a more acurate distribution be if you rerolled 3's and 4's. Since adding 3's and 4's doesn't skew the results towards higher numbers like rerolling 6's does.
full_plate 03-08-04, 12:17 AM	Actuall come to think of it if you really wanted to be accurate and physical, in that there are extreme successes, that there are extreme failures too. In that case reroll all 1's and subtract six and then the number you just rolled. So not only can you kill that red dragon over there but you can kill your best friend right next to him. Just a thought. I wonder what that probability curve would look like, I think tomorrow I am going to program it to find out.
full_plate 03-09-04, 12:48 AM	*bumb*
SirisC 03-09-04, 01:33 AM	I like very much. i ran some numbers on this and about 83% of the time you will roll 1 thru 5 on the d6, 14% of the time you will get 7 thru 11, 2% of the time you get 13 thru 17, .3% of the time you get 19 thru 23, etc.
Tam 03-09-04, 05:06 AM	Hey, Could someone try the following variant : On a 6, rethrow and add 3 to the result [-> average at 3,6 x 3 = 10,8] On a 6, rethrow and add 2 to the result [-> average at 3,4 x 3 = 10,2]
SirisC 03-11-04, 01:06 AM	here are the stats i rolled testing this system out 1st 2nd 3rd 4th 5th 6th 7th 8th 18 27 15 10 26 11 17 27 15 17 19 15 24 15 17 20 33 20 24 13 31 11 17 16 17 22 25 12 17 12 36 15 18 22 20 22 23 28 21 23 26 18 15 14 23 38 20 20 this definitely gives high powered results considering the lowest set of stats has a 22 as the highest and 10 as lowest.
MoonHawk 03-11-04, 01:30 PM	This is intended to replace a d20 roll, not stat rolls. Still, the fact that you couldn't get less than a 10 is pretty disturbing.
SirisC 03-11-04, 07:30 PM	Originally posted by MoonHawk This is intended to replace a d20 roll, not stat rolls. Still, the fact that you couldn't get less than a 10 is pretty disturbing. not really, when rolling 3d6 or 4d6 drop lowest for stats I can remember only 1 time that I rolled a PC with less than 10 in a stat.