Hypothetical Statistics Question

[Tuzenbach]

Let's say that I was dead set upon having a certain spell or weapon or monster hit for 3-11 points of damage. Odd, I know, but this is hypothetical! Would the following methods generate different probability curves, or the same?


2d5+1

2d4+d3

d4+d6+1

4d3-1

d9+2

d7+d3+1


If the probability curves are different, which method would be *most* fair to all parties involved? Thanks!

 

[nhl_1997]

Those systems will have different probability distributions. In general, more dice correspond to more stability around the average (less likely to reach the end points of your desired range.)

What do you mean by "*most* fair to all parties involved"?

 

[Amaroq]

As noted above, all different.

d9 + 2 generates an equal, 1 in 9 chance of rolling each of the numbers you're looking for. The distribution curve is a flat line.

d4 + d6 + 1 generates a flat-capped distribution curve;
3: 1 in 24
4: 2 in 24
5: 3 in 24
6: 4 in 24
7: 4 in 24
8: 4 in 24
9: 3 in 24
10: 2 in 24
11: 1 in 24
In other words, your people are most likely to roll a 6, 7, or 8 (half the time, total), and least likely to roll a 3 or 11 (once in 24 rolls, each)

2d5 + 1 will generate a fairly similar curve, but with a peak at 7:
3: 1 in 25
4: 2 in 25
5: 3 in 25
6: 4 in 25
7: 5 in 25
8: 4 in 25
9: 3 in 25
10: 2 in 25
11: 1 in 25

You can figure out each curve by counting the number of ways in which you can roll any given number; the 'in N' is calculated by multiplying each of the terms after a 'd' in them:
For example, for d4 + d6 + 1, the 'in N' is (4 * 6), or 24.
The '3' in d4 + d6 + 1 can only be acheived if both the d4 and the d6 show 1. That's 'one way' out of (4 * 6) possibilities, or 1 in 24.
The '4' can only be achieved if either [the d6 shows 1 and the d4 shows 2], or [the d4 shows 1 and the d6 shows 2]; that's 2 ways, or 2 in 24.
Etc.

The question 'which is most fair to all parties involved' is fairly meaningless without something to compare it to. If you decide to create two creatures, each doing 3-11 points, and one uses d9 + 2 and the other uses d4+d6+1, is it fair? The former does his 'extreme' results more frequently, while the other does the 'average' results more frequently, but the odds of one doing more damage than the other in any given roll should be the same.

What you're left with is 'what do you prefer'? Would you rather have a large amount of random variance in the utility of this spell or monster? If so, choose d9 + 2. Would you rather have the largest amount of 'predictability'? If so, choose 4d3 - 1. Would you rather have a mixture of the two, with some predictability but a reasonable chance of extreme result? Then 2d5+1 might be 'right' for you. Another concern is 'how easy is it to roll'? Eh, d4+d6+1 is close enough to 2d5+1 for me, plus there's no odd division or rerolls to slow down calculation of the total. I'll just use that.

Of course, our question back to you is, why not use 2-12 and good ol' 2d6?

 

[Patryn of Elvenshae]

The above posters are basically asking if you want your 3-11 to be a Greatsword (2d6) or a Battleaxe (1d12)?

(With the caveat that greatswords can't roll a 1; I know ... )

 

[nhl_1997]

If the probability curves are different, which method would be *most* fair to all parties involved? Thanks!

One thing to consider:
Greater chances in major fluctuations tend to increase PC mortality rate.

Suppose everyone and everything exclusively uses this 3-11 weapon (spell, ability, whatever.) For any specific distribution, there is an equal chance for the dice to favor the PCs or their opponents. Obviously, if the dice favor the opponents, then that tends towards PC death. The PCs go through many encounters, so eventually, the dice will favor the opponents.

For a flat distribution (1d9+2), when the dice favor the opponents, they'll be dealing out well above average damage. For a non-flat distribution (4d3-1 or any other distribution you proposed), the above average dice rolls will correspond to only "slightly above average" damage.

Distributions ranked from most flat to least flat:
1d9+2 (100% flat)
1d7+1d3+1 (flat from 5-9, triangular at tails)
1d4+1d6+1 (flat from 6-8, triangular at tails)
2d5+1 (triangular, same distribution as a greatsword)
2d4+1d3 (peak sharper than triangular)
4d3-1 (sharpest peak)

Over the long-haul, 4d3-1 favors the PCs the most and 1d9+2 favors their opponents the most.

 

[Tuzenbach]

What do you mean by "*most* fair to all parties involved"?

1d9+2 (100% flat)

Ah, you see? Because I'm not of the mathematical mind, I wasn't able to use proper terminology in my question. What I should of said was, "which of these results in an equal probability for all numbers?". Now that I know it's the d9+2, I think I should forget about the 3-11......that is, until I've invented the d9!

Of course, our question back to you is, why not use 2-12 and good ol' 2d6?

That's easy! It's because I refuse to be bound by traditions and conventions. But that 2-12 got me to thinking. This whole time players have only had a 1 in 36 chance of rolling a 12?! Yet, they have a 1 in 6 chance of rolling a 7. I think that sucks.

My question to YOU, is why not just have a 1d10+1? Sure, you lose out on that extra point, but you'll get max. damage with a 1d10+1 around five times more than you would using a 2d6.

Jeez, were these varibales ever even taken into consideration when Gygax (1 in 216 to score max. damage w/ the two-handed sword versus large creatures!) and Cook wrote their respective versions of the rules? I think that since over 90% of the weapons have "flat rates" of damage, they ALL should. Otherwise, what looks like a bonus turns out to be a jip.

 

[Starman]

That's easy! It's because I refuse to be bound by traditions and conventions. But that 2-12 got me to thinking. This whole time players have only had a 1 in 36 chance of rolling a 12?! Yet, they have a 1 in 6 chance of rolling a 7. I think that sucks.

My question to YOU, is why not just have a 1d10+1? Sure, you lose out on that extra point, but you'll get max. damage with a 1d10+1 around five times more than you would using a 2d6.

Jeez, were these varibales ever even taken into consideration when Gygax (1 in 216 to score max. damage w/ the two-handed sword versus large creatures!) and Cook wrote their respective versions of the rules? I think that since over 90% of the weapons have "flat rates" of damage, they ALL should. Otherwise, what looks like a bonus turns out to be a jip.

It's a trade-off. The person rolling 2d6 is going to do average damage more often. They will consistently roll 7 more often than 12 or 2. The person rolling 1d10+1 has just as good a chance of rolling 2 as he does 7 or 11.

Starman

 

[dcollins]

Jeez, were these varibales ever even taken into consideration when Gygax (1 in 216 to score max. damage w/ the two-handed sword versus large creatures!) and Cook wrote their respective versions of the rules?

Absolutely, yes.
AD&D 1st Ed. DMG has a 3d6 bell-curve chart & analysis on p. 10.

D&D 3.0 DMG has a discussion on how increased randomness (like your totally-flat distribution) hurts the PCs more than NPCs on p. 64-65.

 

[nhl_1997]

Slightly more specific for a single encounter, average dice rolls on both sides favors the side expected to win the encounter (since expectations to win are based on average dice rolls from both sides.)

Suppose the PCs are expected to win but it should be a "close-contest". Two options that might swing the advantage towards the opponents:

1) Below average PC dice PLUS average or above average NPC dice
2) Above average NPC dice PLUS average or below average NPC dice

In all distributions (based on dice), the probability of above average rolls equals the probability of below average rolls. The probability of deviating from the average increases as you decrease the physical number of actual dice rolled.

In that context, 1d9+2 help the underdog and the other distributions help the favorite (with 4d3-1 being the most beneficial to the favorite.) If, in your campaign, the PCs are the favorite to win most fights, then flat distributions help their opponents and spiked distributions help the PCs.

By the way, 1d9+2 is fairly easy to emmulate. Just roll 1d10 and reroll a 10. To save time, you could roll two dice designating one to be "primary." If the primary is a 10, then refer to the secondary (and reroll if that is also a 10.)

 

[Kae'Yoss]

That's easy! It's because I refuse to be bound by traditions and conventions.

What do you mean by that? You mean conventions like the d20 rules? Or "dice you can easily obtain, without blowing a lot of money into it?" I'm not a big fan of all that order stuff, but some of it does make sense, when it makes everything easier...