Dice and Odds

[orsal]

I wanted to use the 2d6 + mod for more complex situations where a bell curve distribution may be useful.
...
Since my brain understands percentages much easier than the 2d6 bell curve I asked for that information.

I'm not sure if this matters to you, but you don't get anything like a bell curve with only two dice. You get a triangular shape -- as you move away from the mode (the most probable value) the probabilities drop uniformly. In a bell curve, the probabilities drop off slowly near the mode, more steeply a little further away, and then slowly again toward the edges.

In general, the more dice you have, the closer your distribution gets to a bell curve, but you need at least three dice to get it even slightly curved.

Now, I'm thinking it may be that all you really care about is a modal distribution -- one that has a most probable value, with probabilities dropping off the further you get from the mode. If you don't care whether they drop off linearly (2 dice) or in a curved way (3+ dice), then 2 dice should be fine for you.

 

[Baron Opal]

But wait, there's more...

Next question.

Is the following statement true?

Each step in the following progression will give me consistently higher results.

1d6

1d6+1

2d6, take the single higher roll.

2d6, take the total.

6 + 1d6, take the total.

The real question likes in the difference between 1d6+1 & 2d6, take the higher. If true, at what value of X does 1d6+X a better choice than 2d6, take the higher?

 

[orsal]

Next question.

Is the following statement true?

Each step in the following progression will give me consistently higher results.

Consistently higher? Not sure what you mean. Are you thinking higher on average? On that assumption, let me number your schemes:

(1) 1d6

(2) 1d6+1

(3) 2d6, take the single higher roll.

(4) 2d6, take the total.

(5) 6 + 1d6, take the total.

Trivially, (1)<(2) (adding 1 always raises a number), (3)<(4) (since 1 can never be higher than the other die and is often lower), and (4)<(5) (since replacing one die with an automatic 6 usually raises, and never lowers, the total). The only nontrivial part is whether (2)<(3).

roll    prob(1d6+1)    prob(higher of 2dice)
1        0             0.0278
2        0.1667        0.0833
3        0.1667        0.1389
4        0.1667        0.1944
5        0.1667        0.25
6        0.1667        0.3056
7        0.1667        0
average  4.5           4.47

So, it turns out, on average 1d6+1 gives you slightly higher than the better of two dice.

Whether slightly higher means better, depends on what you want. If you have a particular target you'd like to reach, and care more whether you hit that level than how much above or below it you get, then 1d6+1 gives you a better shot at getting at least 2 (certain for 1d6+1 but not for the better of two dice), 7 (possible only with 1d6+1), or 6, while taking the better of two dice gives you a better chance of getting at least 3, 4, or 5.

The real question likes in the difference between 1d6+1 & 2d6, take the higher. If true, at what value of X does 1d6+X a better choice than 2d6, take the higher?

X=1, if you want the highest average result.

 

[Baron Opal]

Drat. That's what I thought but I wan't sure if the break point would be +1 or +2. It forces me towards a simpler solution, which since I'm running b/x is the point I guess.

 

[Baron Opal]

Drat. That's what I thought but I wan't sure if the break point would be +1 or +2. It forces me towards a simpler solution, which since I'm running b/x is the point I guess.

Thanks for the timely response.

 

[Baron Opal]

I've started a thread with my thoughts for bardic charm in a new thread.

Regarding the choice between 1d6+1 vs 2d6 t.h., I presume that scales with the die. If the former is close, then 1d12+2 is close to 2d12 t.h., and 1d20+4 is close to 2d20 t.h.?

 

[orsal]

I've started a thread with my thoughts for bardic charm in a new thread.

Regarding the choice between 1d6+1 vs 2d6 t.h., I presume that scales with the die. If the former is close, then 1d12+2 is close to 2d12 t.h., and 1d20+4 is close to 2d20 t.h.?

Pretty much. With dn's, on average taking the higher of two dice beats a single die by just less than n/6. So yes, 1d12+2 is marginally better than the higher of two d12s, and 1d20+3<(higher of 2 d20)< 1d20+4.

 

[Baron Opal]

That's funny. I would always have chosen two dice over one die with a bonus. However, that's not the same as a re-roll is it? I get the sense that the halfling racial power in 4e is a bit better still.

 

[orsal]

That's funny. I would always have chosen two dice over one die with a bonus. However, that's not the same as a re-roll is it? I get the sense that the halfling racial power in 4e is a bit better still.

Rolling two dice and taking the higher can be thought of as one die, with a reroll, with the added option to revert to the first roll if the second one turns out to be lower. So it's better than an optional reroll. However, if it's a hit-or-miss roll (i.e. one where all rolls less than N have the same effect, and all rolls greater than or equal to N have the same effect), and you know whether it succeeds before deciding whether to reroll, then the two are equivalent.