Help me understand "average rolls"

[Desdichado]

1) When selecting a DC it is very simply to calculat success ratio. (21 - Roll needed) * 5 = % chance of success. Makes designing adventures easy.

Designing adventures is easy already without running any formulae. I don't understand what this has to do with using d20 or 2d10, however. I just eyeball challenges.

2) Drastic alteration to game balance. The rules were deisned with a d20 in mind not 2d10s when you use 2d10s you are changing the entire balance scenrio of the game. Also you could be doing away with critical failure and if you do not then you affect game balance even more drastically. Let's take a look at it from a statistical point of view.

There is no critical failure in the rules now, so I don't see how this changes anything. Also, the design inherent in the game should be more or less unaffected, as the range of possible results is still pretty much the same. You can still do it just fine, as long as you accept that the results are more like a bell-curve rather than a flat-line curve.

When you roll two dice and add them together you begin to represent a bell curve. This anamoly happens because of the number of combinations that can make any given number.

No kidding! That's the whole reason to switch from d20 to 2d10 -- bell curves are desirable relative to flat-line curves, as far as I'm concerned.

When you take the set of possible results from adding 2d10 together it is 20, if you take the roll of 0,1 as a 1 and the roll of 0,0 as a 20. We will assume this for all future calculations since it is accepted practice to make 00 = 100 and 01 = 1 when using 2d10 for precentile rolls. You have 100 (10*10) possible rolls and only 20 possible out comes. However the number of combinations that make any given number are not equal which is where the bell curve representation comes from.

Bad assumptions. The range changes from 1-20 to 2-20 as far as I'm concered.

The anomoly at 11 happens because we steal 2 11s to make 1s. So the odds of beating a given DC become very difficult to calculate and therefor a fair challenge becomes difficult to design. The benefit gained is very small for the complexity added therefor sticking with 1d20 is a better alternative then 2d10.[/color]

Well, why make things difficult, then? Just go with the more accepted practice of all 0s = 10 and all 1s = 1? Don't call the system complicated because you go out of your way to make it be so.

 

[buzz]

Bad assumptions. The range changes from 1-20 to 2-20 as far as I'm concered.

Indeed, the curve is very slight. The chance of rolling 10 or 11 goes from 5% up to about 8%. The biggest difference the 2d10 introduces is that criticals and "fumbles" (i.e.,20s and 1s) become less frequent, though not drastically so. This may be prefereable to some, undesirable to others.

Still, I agree that figuring out probailities on the fly is easier (for me) with a flat distribution.

Part of me wonders if this is the reason GoO decided to go with 2d10 for SAS Tri-Stat, assuming they knew they were also going to do a d20 version. It makes the two games more similar than if they had gone with BESM's 2d6 or even a HERO-esque 3d6.

 

[shurai]

*literally just walked out of his Probability I final*

*reads the boards*

AHH! Get it away! Away! *runs off screaming*

Seriously though, what you're talking about when you find the 'average result' of a die roll is to find what's called the Expected Value. Basically the Expected Value is equal to the following, where X is a random variable, and P(X = x) is the probability that the random variable X will take on the value x:

E(X) = sum over all possible results for X of [ x * P(X = x) ]

So the expected value of a single d20 roll is equal to:

E(X) = sum{1-20: x}[ x * P( X = x) ] = 0.05(1) + 0.05(2) + 0.05(3) + . . . + 0.05(20) = or the well-known 10.5.

If you want to know what's really useful about averages etc is measuring how far X will wander from its expected value. In probability this is the Variance of X, Var(X) and from it we get the Standard Deviation.

Var(X) = E[ (X - E(X) )^2 ]

sd(X) = sqrt( Var(X) )

Which can gives and indication of how far a random variable will wander from its expected value.

-S

 

[Desdichado]

Still, I agree that figuring out probailities on the fly is easier (for me) with a flat distribution.

Oh, no doubt. But do you really need to figure probabilities? I just -- as I said earlier -- eyeball challenges. It's close enough for me.

 

[shurai]

Indeed, the curve is very slight. The chance of rolling 10 or 11 goes from 5% up to about 8%. The biggest difference the 2d10 introduces is that criticals and "fumbles" (i.e.,20s and 1s) become less frequent, though not drastically so. This may be prefereable to some, undesirable to others.

Foist of all, the chances of rolling an 11 on 2d10 isn't 0.08 but 0.1, and the chances of rolling a 10 on 2d10 is 0.09. Also, I'm no sure what your definition of 'drastically' is, but the probability of rolling minimally falls from 0.05 to 0.01. For most people that would be a pretty drastic drop.

The curve isn't that slight either, at least not to me (though it's a subjective thing so maybe we just disagree about what 'slight' is).

Anyway, one thing I can be sure of is the math, so here it is:

Sure, for getting an 11 it goes up by 5 percentage points, but if you look at a spread of values, things get more pronounced.

Let's look at the chances of getting a range of values close to the expected value of both rolls, say 8 to 13. For the d20, there's 6 values that equal success, all equally likely, and 20 possible outcomes, so 6/20 or 0.3.

For 2d10 it's a bit more complicated. There are 100 possible rolls, all equally likely. Several combinations of rolls will result in any given target result, as follows:

8: 17, 26, 35, 44, 53, 62, 71
9: 18, 27, 36, 45, 54, 63, 72, 81
10: 19, 28, 37, 46, 55, 64, 73, 82, 91
11: 10, 29, 38, 47, 56, 65, 74, 83, 92, 01
12: 20, 39, 48, 57, 66, 75, 84, 93, 02
13: yay symmetry! same as 9.

So the total number of outcomes that fall in this range is 7+8+9+10+9+8 or 51.

So in a d20 roll, the chances of the roll falling in the range of 8 to 13 are about 3 in 10, but the same range for 2d10 is more than 5 in 10.

EDIT: Holy convenient math error, batman!

 

[buzz]

The curve isn't that slight either, at least not to me (though it's a subjective thing so maybe we just disagree about what 'slight' is).

I think we do. I'm thinking of a 3d6 curve (probably the next most common die mechanic), where the difference between extremes is *much* more drastic (3s and 18's at .5%, 10s and 11s at 12.5%).

 

[Terraism]

I think we do. I'm thinking of a 3d6 curve (probably the next most common die mechanic), where the difference between extremes is *much* more drastic (3s and 18's at .5%, 10s and 11s at 12.5%).

That's why rolling stats is so much fun.

 

[buzz]

The Sage weighs in

Just FYI:

In a message dated 12/8/02 3:08:47 PM, buzz@enteract.com writes:


<< Hello,


On p.58 of the PHB and p.13 of the DMG, some mention is made of "an
average roll" on 1d20. The PHB quote says something like:


"...on average, Devis will roll a 10 or 11..."


What I'm curious about is this: are the designers talking about the
arithmetic mean here or a frequency distribution?<<


It's the former.


>>In the former case, I take it that the phrase should be understood as
meaning: "on average, 50% of Devis' rolls will be above 10 and 50% will
be below; knowing this, we can assume blah blah blah..."<<


Correct.

 

[shadowlight]

Re: The Sage weighs in

Just FYI:
...
>>In the former case, I take it that the phrase should be understood as
meaning: "on average, 50% of Devis' rolls will be above 10 and 50% will
be below; knowing this, we can assume blah blah blah..."<<


Correct.

Also just FYI, the correct statistical way of saying this is that the long run average roll will be a 10.5 (and just to be nitpicky, if you want to use the 50% way of saying it, it would be "50% of Devis's rolls will be above 10 and 50% will be 10 or below").

"On average" (as in "...on average, Devis will roll a 10 or 11...") means that if you were to add up all the d20 rolls you ever made and divide that by the number of rolls, you'd get between 10 and 11...

 

[mmu1]

At least one thing that'd need a complete overhaul if you changed from a d20 to 2d10 would be the critical hit system. All of a sudden, weapons that crit on a 20 only crit 1/3 as often as weapons that crit on a 19-20, and, what, 1/6 as often as 18-20 weapons? Things like Keen Edge and Improved Critical couldn't simply double the threat range anymore, either.