What is the value of rolling two d20s and keeping the best result?

[harpy]

I've been googling, searching forums, etc. I just can't seem to find this issue being address directly, but it's bound to have come up in the past...

What is the "value" of rolling two d20's and keep the better result?

That is, how do the above stack up versus strait bonuses to the roll?

There is the bell curve factor, or the option of "taking 10", but in both of these situations you aren't being bound necessarily by the average result.

In 3.5 I was playing a Gnome Archivist in one game that had the Trivial Knowledge feat which allows rolling two d20s and keeping the highest result on knowledge checks. With optimized knowledge skills, and using it for the "Dark Knowledge" feature of the archivist, it was... a bit cheesy. I'd lazily pick up my two d20s, and after many many sessions of making Dark Knowledge checks (Which add insight bonuses to the party to fight monsters) I only had one result which made me deal out the minimum insight bonus for an encounter.

So in that situation having the "roll two d20s and keep the best" was really valuable.

I guess I'm trying to tie down a metric value as to how valuable that kind of rule would be in a d20 game in general. I know that it is a moving target, since the bonuses and target numbers are all over the place. In general it seems like it shifting towards a bell curve result, however you aren't bound by the bell curve, so it's both a safety net and a spring board all in one.

But in a fuzzy, inchoate way is it "kinda-like" having a +5 bonus? A +10 bonus? My math fails me!

 

[Phaezen]

The average for a D20 is 10.5, the average for Rolling Twuce, taking the highest is 13.825.

However the maximum value (20) and the minimum value(1) are still the same, so you can't really equate it to a straight bonus.

 

[Nadaka]

Its the equivelant of a +3.325 bonus from a purely mathematical POV.

What you do is assume a base line of rolling 1d20.
For each of the possible results of that first d20 and each of the possible results of the second d20: record the difference if the second d20 is higher.
Then divide the sum of those differences by the number of combinations (20*20).

However, in practice it is much more effective than that when the target you have to roll on the d20 is low.

 

[drothgery]

(deleted; better explanations with more math above)

 

[FireLance]

From a previous thread on the topic, the effective bonus for a double roll works out as follows:

[Note: RTH = Number required to hit on a d20, CTH = chance to hit, RRCTH = chance to hit with a reroll, EFFB = effective bonus (rounded)]

RTH: 2; CTH: 95%; RRCTH: 99.75%; EFFB: +1
RTH: 3; CTH: 90%; RRCTH: 99.00%; EFFB: +2
RTH: 4; CTH: 85%; RRCTH: 97.75%; EFFB: +3
RTH: 5; CTH: 80%; RRCTH: 96.00%; EFFB: +3
RTH: 6; CTH: 75%; RRCTH: 93.75%; EFFB: +4
RTH: 7; CTH: 70%; RRCTH: 91.00%; EFFB: +4
RTH: 8; CTH: 65%; RRCTH: 87.75%; EFFB: +5
RTH: 9; CTH: 60%; RRCTH: 84.00%; EFFB: +5
RTH: 10; CTH: 55%; RRCTH: 79.75%; EFFB: +5
RTH: 11; CTH: 50%; RRCTH: 75.00%; EFFB: +5
RTH: 12; CTH: 45%; RRCTH: 69.75%; EFFB: +5
RTH: 13; CTH: 40%; RRCTH: 64.00%; EFFB: +5
RTH: 14; CTH: 35%; RRCTH: 57.75%; EFFB: +5
RTH: 15; CTH: 30%; RRCTH: 51.00%; EFFB: +4
RTH: 16; CTH: 25%; RRCTH: 43.75%; EFFB: +4
RTH: 17; CTH: 20%; RRCTH: 36.00%; EFFB: +3
RTH: 18; CTH: 15%; RRCTH: 27.75%; EFFB: +3
RTH: 19; CTH: 10%; RRCTH: 19.00%; EFFB: +2
RTH: 20; CTH: 5%; RRCTH: 9.75%; EFFB: +1

 

[Obryn]

Also, and IMO most importantly, your chances of critting are just shy of 10%, and your chances of automatically missing are 1/400.

In other words, the averages don't tell the whole story. Whenever you roll two dice and keep the better, you're creating a curve that's skewed towards higher numbers, not just changing your average roll.

EDIT: Wow, I waited too long between hitting Reply and hitting Post!

-O

 

[Celebrim]

The average for a D20 is 10.5, the average for Rolling Twice, taking the highest is 13.825.

However the maximum value (20) and the minimum value(1) are still the same, so you can't really equate it to a straight bonus.

True, however you can say that the odds of producing the lowest possible result when rolling two dice and keeping the best result are 1:400 (as opposed to 1:20), and the odds of producing the best possible result are 39:400 (as opposed to 1:20). That means that you almost double the chance of a 20 and reduce the chance of a 1 to a very small number. Since '20' and '1' have special significance at certain times in the game, this means that a reroll like that is very valuable at those times compared to a linear bonus. Consequently, we can say that the value of a reroll is probably greater than a +4 bonus most of the time.

 

[vagabundo]

The value - to a player - is the extra quantum of joy they get in rolling an extra die...

Seriously, I know players who pick these things for that trill..

 

[harpy]

The value - to a player - is the extra quantum of joy they get in rolling an extra die...

Seriously, I know players who pick these things for that trill..

Heh... yes, this might be the greatest value

Wow! Thanks for all of the replies, very helpful.

The Anydice suggestion (AnyDice Calculator) is also great for those who work better with visuals and charts, like myself.

So in a vague way it is like having a +3 to +5 equivalent to the roll depending on the target value, and it greatly reduces "1" results and increases the likelyhood of "20" results.

So in terms of the d20 system, giving a "roll two keep best" ability for attack rolls is a bit much in game balance terms. If you were to use this as an option it would be best for "soft power" abilities that don't have a great deal of risk in failure, such as a lot of skill checks.

Most of all of these very helpful replies I've been reading have been focusing on targets within 1-20, but what if the target goes above 20? From what has been said so far as you go towards the extremes the value tends to diminish.

What if you have rolls where the target is 25, 30, or 35? You'd have to assume that the roll would have a bonus of +5, +10 or +15 so that it is possible to reach the target. Does having those new targets and bonuses simply slide the entire scale along?

Example, if you have a target of 25, and you roll two d20s with +5 to each roll and then pick the best, are you just shifting that fuzzy "+3 to +5" value up five notches, or are other wonky effects kicking in because you've gone past the number of faces on the die?

 

[pawsplay]

It's almost like being able to take 10, but instead of not critting, you crit nearly twice as often. In general, this should be reserved for noncombat abilities or abilities with very limited usage.