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View Full Version : Effects of Bell-Curve Mechanic [Dice Probabilities]



kieza
2015-04-11, 03:21 PM
So, I'm working on a system which uses "2d10, roll to match or beat a target number" as its core mechanic. I'm also using a mechanic based on 5e's advantage and disadvantage:

For every advantage (flanking, surprise, higher ground, etc.) you have on a core roll, you roll one extra d10, and discard the lowest. So, with one advantage, you roll 3d10 and drop the lowest. With two advantages, you roll 4d10 and drop the two lowest.

Disadvantage (attacking someone with cover, attacking while prone, etc.) works the opposite way. For every disadvantage, you roll one extra d10 and discard the highest.

Now, this combination does three things that I like:
--2d10 gives you a bell curve, so that average rolls are more probable than very high or very low rolls. This makes skill a larger factor than chance in most circumstances.
--It provides a simple mechanic for handling an advantage or disadvantage: this means less adding up bonuses, and keeps play from bogging down as you add up a half-dozen small, situational modifiers.
--Everyone likes rolling lots of dice.
--And, since I'm using 2d10 as the core mechanic, "roll one more d10, discard highest/lowest" works much better than "roll 2d10 twice, pick the higher/lower result." (I considered copying 5e's advantage/disadvantage completely, but then I realized that you couldn't roll 4d10 at once unless you had your d10's in color-coded pairs.)

But, I've found one frustrating side effect:
If you have advantage and disadvantage on the same roll, meaning that you roll 4d10 and discard the highest and lowest dice, the probability distribution has the same average as an unmodified roll (2d10). This is good--the advantage and disadvantage cancel each other out that way. But--if you have advantage and disadvantage, it makes the standard deviation of the distribution smaller, meaning that very high and very low rolls are less likely. In particular, it makes a critical hit (natural 18-20) or critical failure (natural 2-4) far less likely.

So, my questions to the playground:
1) Is this a problem at all? As players or DMs, would you find it strange/bad that when you have equal advantage and disadvantage, your rolls have the same average as if you had no advantage/disadvantage, but you're unlikely to get a critical hit/failure?
2) Would it be a better idea for advantage and disadvantage to just cancel each other out? That is, if you have two advantages and one disadvantage, should you just roll 3d10 and drop the lowest, like you had only one advantage?
3) If you think it's a problem and you think that cancelling advantage and disadvantage is a bad idea, can you suggest an alternative critical hit/failure mechanic? I'd prefer something where advantage makes critical hits more likely, disadvantage makes critical hits less likely, and having an equal number of advantages and disadvantages doesn't change critical hit chance.

Oh, and for those interested, here's the Anydice code I was using to examine this mechanic:
output [highest 1 of 2d20]
output 2d10
output [highest 2 of 3d10]
output [highest 2 of 4d10]
output [lowest 2 of 3d10]
output [lowest 2 of 4d10]
output [middle 2 of 4d10]

Khedrac
2015-04-11, 03:42 PM
I think it would make sense for the disadvantage effect to be "remove one dice from the pool" rather than "drop highest" - i.e. option 2.
This is a lot simpler to roll.

Also this way if there are too many factors against you it changes from "nearly impossible" (drop highest) to "impossible" (not enough dice left to achieve result) - this is not necessarily a good side effect.

Also you state that "Everyone likes rolling lots of dice" - I find that this is not always true. Those who are less mathematically able and those who don't own enough of the relevant dice often prefer to roll fewer. Likewise people not gaming at a table - if you are rolling on a small flat surface the more dice the more hassle.

Final one for fewer dice is how people perceive their luck. For most things my luck is streaky, but I reckon I tend to be poor at rolling stats (why I like point buy) but the system where my luck really is poor is OWoD where the more dice I roll the more I can mess up (I once rolled 7 or 8 dice for a difficulty 7 check - I think 3 fumbles was the net result, it might have been more). As a result in OWoD-type systems I actually prefer rolling fewer dice - I cannot fail as badly! In your proposed system I would expect the "discard high" mechanic to hammer my results and the "drop a dice" mechanic not to be too bad.

Mastikator
2015-04-11, 04:30 PM
1. Sounds great. Crits and fumbles are the bane of my experience, anything that makes them less likely and less extreme is good in my opinion.
2. That would make things easier, which is good. The less of a hassle it is to play the better.
3. A simpler option is just 2d10 + circumstance modifier number + base modifier number VS difficulty check number. Honestly, changing it so that getting 10,10 on 2d10 just gives +1d10, and 1,1 on 2d10 takes away -1d10 value. And anything WAY above DC means "critical success" and anything way below DC means critical failure.

The Evil DM
2015-04-11, 04:33 PM
2d10 does not provide a bell curve.

It provides a tent curve with the following probabilities.

2 = 1%
3 = 2%
4 = 3%
5 = 4%
6 = 5%
7 = 6%
8 = 7%
9 = 8%
10 = 9%
11 = 10%
12 = 9%
13 = 8%
14 = 7%
15 = 6%
16 = 5%
17 = 4%
18 = 3%
19 = 2%
20 = 1%

The sides of the tent are linear peaking at total of 11. This does not invalidate your plan, but you require at least three dice to get a bell curve distribution.

I have extensive experience and many existing spread sheets evaluating dice probabilities. PM me an email address I can forward them to you.

Feddlefew
2015-04-11, 05:31 PM
3d6 makes a nice bell curve.

If you want it to fit into a 1-20 range, I've read that using 3d6 and two fudge dice* makes a REALLY nice bell curve. Of course, that means you're rolling (and summing) 5 dice instead of 1 die, so you need to ask yourself if that method is right for your game.


*Fudge dice are six-sided dice with two "+" sides, two blank or "0" sides, and two "-" sides, ICYDNK.

kieza
2015-04-11, 07:48 PM
I think that there might have been a little misunderstanding, so let me try to clear this up...

The core roll is 2d10, in the same way that the core roll for the d20 system is 1d20. You roll 2d10's, and add a relevant modifer, then check if you beat the target number. The advantage/disadvantage system is just meant to cut down on situational modifiers: when you're flanking a target that took Total Defense and one of your allies is aiding your attack, it's "Roll 5 dice, discard the bottom 2 and top 1, and then add your modifier" instead of "Roll 2 dice, add your modifier, add 2 for flanking, add 2 for aid another, subtract 5 for improved cover." You only ever have to add 2 dice and a modifier, and the modifiers seldom get above +10 even at high levels.

Also, the range of the dice is always 2-20, no matter how many advantages/disadvantages you get, because advantage and disadvantage cause you to roll 1 extra die and then discard 1 die, leaving you with 2d10. Advantage can't allow you to do something you wouldn't normally be able to, and disadvantage can't prevent you from doing something you normally could. They can only affect the probability of success or failure.

I'm calibrating the the target numbers so that a character can be successful at something they're moderately good at on a roll of natural 11. That's a 55% chance on a 2d10 roll with no advantage or disadvantage; with 1 advantage it increases to 78.5%, and with 1 disadvantage it falls to 29%.

If the task is more difficult, requiring a natural 16, then there's a 15% chance of success on a normal roll. With advantage, it's a 33% chance and with disadvantage, it's a 4% chance.

Finally: I'm aware that 2d10 does not produce a classic bell-shaped curve. That's because it is not a Gaussian function. It does have two defining traits of the classic normal (bell-shaped) curve: its most likely outcome is also the mean and median of the distribution, and the probability of an outcome strictly decreases as the distance of the outcome from the mean increases. That's why I called it a bell-shaped curve. (Also, if you roll larger numbers of dice, like 4d10 or 5d10, it is a good way to quickly approximate a normal distribution; maximum error when comparing a 5d10 distribution to a normal distribution with mean 17.5 and stdev 3.819 is 5%.)

Coventry
2015-04-11, 10:43 PM
I am in favor of the one for one cancellation, specifically because of the "roll greater than 15" scenario you described. A person with 10 advantages and 10 disadvantages should be as capable as someone with no advantages or disadvantages, but because of the centering of the dice when rolling 22 of them, his real odds of hitting a 15 are really low.

Coventry
2015-04-11, 10:53 PM
Also ... I suspect that the guy with 10 Advantages and 10 Disadvantages will score a 15+ result less often than a guy with 0 Advantages and 1 Disadvantage, which would be contrary to the intent of the design.

I just don't have the brain power at the moment to run the numbers to see if my suspicion is true.

kieza
2015-04-11, 11:11 PM
Also ... I suspect that the guy with 10 Advantages and 10 Disadvantages will score a 15+ result less often than a guy with 0 Advantages and 1 Disadvantage, which would be contrary to the intent of the design.


Anydice agrees with you.

NichG
2015-04-11, 11:21 PM
If this is based on 5e design, advantages and disadvantages don't stack, so there's no situation where you'd have a guy with 15 advantages and disadvantages.

Generally for this kind of dice design question, I find that the real question to ask is 'are the distinctions noticeable by the players, or only because you're plotting the probability distribution'. When the math is really obvious, like with a d20, players will claim to notice 5% changes in the success rate, but if that change in success rate were obfuscated or invisible somehow, I wonder if they'd really be able to tell.

This feels like the kind of thing one would want to run a couple of experiments to test, though the experiment design is a bit tricky since you want to replicate the emotional charge of the situation and also the non-simultaneity of the results (if the players can see 8 successes and 3 failures versus 7 successes and 4 failures, they might estimate probabilities differently than if they experience those events one at a time).