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

[Cadfan]

Firelance's chart is basically correct.

The only thing I'd add, in terms of how to interpret the numbers, is that "effective bonus to hit" isn't the only metric. You could also consider things in terms of the relative increase in your chance of success that you get when rolling twice. Under that view, while you may only have an "effective bonus to hit" of 2 when you need a 19 or better to succeed, you have nearly a doubling of your chances of hitting.

That's basically what the "effective bonus to hit" is doing. Its taking the relative increase in success chance, and figuring out what bonus would give you the same relative increase. But that disguises that in some cases a +2 is better than in others, relative to your original situation.

Eh, its all subjective and interpretive at this point. But I prefer to look at things in terms of relative increase. Once you get used to it, it requires less mental gymnastics.

 

[Cadfan]

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?

What matters is what number you need to roll on the die.

If you are rolling a +0 attack versus an AC of 15, and later are rolling a +5 attack versus an AC of 20, the benefit of rolling twice doesn't change. In both cases you need to roll a 15 on the die, and rolling twice affects your chance of rolling a 15 on the die.

As for your earlier comment on whether rolling twice is "a bit much," well, its all relative, isn't it? Depends what you're charging the character for the ability. In 3e, I'd be a little leery of giving that out to just anyone, because you just know you'll end up with a hasted ranger rolling eight attacks per round with two d20s per roll. But I might consider it if for a character who paid something significant for it- like getting to roll twice on his primary attack by sacrificing his secondary attacks. Assuming he's not playing a supercharger build or something where he only attacks once per round anyways. Its all situational.

 

[Celebrim]

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?

The higher the target, the more relatively valueable a linear bonus is. To be honest, the math here surprised me and invalidates my early intuitive statement. Of course, because '20' has special meaning, things get even more wonky than usual.

Suppose the target is 25, and you have two attackers, one with a +9 bonus and another with +5 bonus and a reroll.

The one with the +5 bonus needs a 20, which he gets 39 times in 400 or 9.75%.

The one with the +8 bonus needs a 16, which he gets 25% of the time. Clearly, the linear bonus is much better here. In fact, the reroll is actually in this case worth a little less than a +1 bonus.

Suppose the target is 10, then the one with the reroll hits it 93.75% of the time, and the one with the greater bonus hits it 95% of the time. In this case, the reroll is worth just slightly less than +4 bonus.

The upshot of this is that you really only want a reroll when the odds of failure are long. Rerolls of good saving throws are great. Rerolls of ordinary skill checks are not so valuable. Rerolls of attack rolls are really valuable mainly if you have a larger than normal chance of a critical, but you probably wouldn't sacrifice a larger linear bonus to get one.

 

[harpy]

So I'm just talking out loud at the moment...


So in 3.5 the progression for a well optimized character for skill bonuses (taking into account full ranks, ability score, skill focus, misc bonus and various item bonuses) would be:

Level 1 = 12
Level 2 = 13
Level 3 = 14
Level 4 = 15
Level 5 = 16
Level 6 = 17
Level 7 = 18
Level 8 = 21
Level 9 = 22
Level 10 = 23
Level 11 = 24
Level 12 = 25
Level 13 = 26
Level 14 = 28
Level 15 = 29
Level 16 = 31
Level 17 = 33
Level 18 = 35
Level 19 = 36
Level 20 = 38

So in looking that over at level 8 you've already got a "take 20" automatically in effect. At level 11 DC 25s are cleared and at level 15 DC 30s are cleared.

If you added a "roll twice, take best" effect then you'd shift all of these by 3 to 5. At low levels it could mean that by level 5 a highly optimized character could be getting a "take 20" effect on their choice skill. It also could mean that in some instances 3rd level characters would be getting up to a equivalent "take 20" in the right situations.

So the overall effect for an optimized skill with roll twice added on is that almost over the entire course of the characters career the skill is going to be trivial to accomplish, save for spotting invisible characters or having to do opposed rolls with other optimized or powerful opponents.

So I guess that is leading to the next question, how does the roll twice, pick best ability affect opposed rolls?

If the opponent doesn't have the ability then we are looking at three d20 rolls being made. And from those charts it comes out to a +5 bonus, though now my illogical mind begins to implode, since the opposition rolls might be making things get really weird. Is it really as simple as "roll three dice and pick the best one" or doe the fact that one player is getting two chances versus the one chance from the other player add loops to the formula?

 

[WalterKovacs]

Odds of rolling X or higher by rolling 2 d20 and taking the better result (1 d20):

20: 9.75% (5%)
19: 19% (10%)
18: 27.75% (15%)
17: 36% (20%)
16: 43.75% (25%)
15: 51% (30%)
14: 57.75% (35%)
13: 64% (40%)
12: 69.75% (45%)
11: 75% (50%)

The left side increases by a diminishing ammount (odds of rolling a specific number X on the 2d20s is 2X-1/400, while the odds on a single d20 it is always 1/20 or 5%. Once you get below 11, you get less than 5% per number on 2d20, so the left side will not increase as quickly as the left side, and they will get closer to each other until you reach the bottom where they will both always roll 1 or higher.

The last number is a big way of showing the increase. If your target number is 11, for example, you go from hiting half the time, to three quarters of the time ... effectively turning half of the misses into hits. [for skills, replace hits/misses with success/failure].

The benefit depends on the goal. Using multiple rolls that exist in 4e, here are some examples:

2 roll for an attack: This is used as an alternative for boosting damage. In this case you are looking for (a) a target number or greater and (b) the chance of scoring a critical hit. This improves the chances of both by a large ammount. For numbers that aren't extremely high, it's close to an effective +4 to hit in terms of increasing the odds of hitting, and close to a 19-20 crit range.

2 roll for a skill check: Ahtletics, is one example of an ability that provides this case. Here you may have a specific goal in mind (i.e. DC for climbing, minimum distance you need to jump), and you will increase the chances of rolling over that ammount. Also, i the case of jumping distance, you may just want to go as far as possible. While you can't increase your maximum distance, you are more likely to roll a high number and go further.

2 rolls for initiative. In this case you don't have a DC, you have competing rolls. Here you just want to roll as high as possible. Again, you can't improve your max roll by having an extra d20, but you will avoid getting bad initiatives (odds of getting 6 or less is less than 10%) while routinely rolling high).

In general, it's similar to taking 10. There is still some risk you'll get worse than 10 (20.25% chance) but you also have a very good chance of getting better than 10 (75% chance).

 

[Umbran]

So in looking that over at level 8 you've already got a "take 20" automatically in effect.

Um, no. Because in Take 10 and Take 20, you add 10 or 20 to the skill, not "assume the total result is 10 or 20."

So, at level 8, you can reliably hit a DC 30, and when you do take 20, you can hit a DC of 40. That is not the same as "take 20 automatically".

 

[Stuntman]

So I guess that is leading to the next question, how does the roll twice, pick best ability affect opposed rolls?

If the opponent doesn't have the ability then we are looking at three d20 rolls being made. And from those charts it comes out to a +5 bonus, though now my illogical mind begins to implode, since the opposition rolls might be making things get really weird. Is it really as simple as "roll three dice and pick the best one" or doe the fact that one player is getting two chances versus the one chance from the other player add loops to the formula?

I did an analysis in the 4E rules forum a few weeks back comparing Improved Initiative (+4 bonus) to Danger Sense (roll twice pick best). Initiative is basically an opposed roll. To summarise, rolling twice falls somewhere between a straight +3 and +4 bonus to your roll. If you have the option to choose a +3 (or lower) bonus or rolling twice, you should choose rolling twice. If you have the option of choosing a +4 (or higher) bonus or rolling twice, choose the +4 (or higher) bonus.

 

[Thanael]

Wow guys, this anydice site is cool.

Compare the results for 1d20h, 2d20h, 3d20h, ... (The h stand for keep the highest value). The first result table lists the frequency of each result from 1 to 20. With 1d20 it is even, all results are equally probably, 2d20 is linearly skewed towards the high higher results, with 3d20h the it gets even more cured with higher results becoming even more probably in comparison to the lower ones.

 

[Mustrum_Ridcully]

This kind of question crops up occasionally. I think we need a FAQ and an easy way to find it. A "FAQ" alone doesn't mean anything if people don't find it or do not check it. [/META]

 

[CubeKnight]

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

Wow, that site is good *bookmarks*
XP for you!