Double attack roll formula

[Flipguarder]

Does anyone have either a table or a formula for finding the theoretical bonus a reroll d20 gives for various to-hits?

For example say I have to roll an 11 to hit, I would like to be able to figure out by either plugging in a formula or looking at a table, how much of a bonus to hit I would get for a reroll.

any of this making sense?

[Nifft]

There is no such formula, because getting a re-roll is a non-linear bonus. It's frequently better than a linear bonus -- particularly in D&D, where no linear bonus would ever save you from a natural 1.

You can work out some interesting results for yourself, by making a 20-by-20 table and coloring in the squares that would be a "hit" under various conditions.

One simple result is: crits (on a natural 20) are almost twice as likely, rising from 5% of outcomes (1-in-20) to 9.75% of outcomes (39-in-400). To visualize this, consider your 20-by-20 grid, and color all cells in the '20' row, and all cells in the '20' column. You'll have one cell of overlap, for a total of (20 + 20 - 1) = 39 colored cells, out of 400 cells.

Cheers, -- N

[Flipguarder]

Well the formula doesn't have to be linear. I guess Ill try to see what I can do if nobody else has done the legwork.

[shroom1000]

I made a formula for the chances of success. It does not have any logic for natural 1s or natural 20s, just the chance of hitting.

1d20 = (21-n)/20
2d20 = 1-((n-1)/20)^2
(n is the roll needed to hit)

Here is the table of odds below:

ToHit   1d20     2d20
 1      100.0%   100.00%
 2      95.0%    99.75%
 3      90.0%    99.00%
 4      85.0%    97.75%
 5      80.0%    96.00%
 6      75.0%    93.75%
 7      70.0%    91.00%
 8      65.0%    87.75%
 9      60.0%    84.00%
 10     55.0%    79.75%
 11     50.0%    75.00%
 12     45.0%    69.75%
 13     40.0%    64.00%
 14     35.0%    57.75%
 15     30.0%    51.00%
 16     25.0%    43.75%
 17     20.0%    36.00%
 18     15.0%    27.75%
 19     10.0%    19.00%
 20     5.0%      9.75%

[FireLance]

Assuming you mean roll twice and take the higher number, the effective bonus 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

[FrozenChrono]

the problem is it's not exact, it varies a lot by target number and there's no way to quantify it at low fail rates.

. . . but on average it''ll increase your rate of success as if you had about a +6

here are the %'s of failure at each target number with 2d20's

1=.25%
2=1%
3=2.25%
4=4%
5=6.25%
6=9%
7=12.25%
8=16%
9=20.25%
10=25%
11=30.25%
12=36%
13=42.25%
14=49%
15=56.25%
16=64%
17=72.25%
18=81%
19=90.25%
20=100%

so if you fail on a 10 or below (or succeed on an 11 or above) you have a 25% chance of failure or a 75% chance of success.

to give you some perspective chances of failure with 1d20 are

1=5%
2=10%
3=15%
. . .
10=50%
. . .
15= 75%
. . .
19=95%

. . . I started posting before anything else was up . . . calculated them manually too. That's annoying.

[Flipguarder]

I know theres going to be a BILLION exceptions to this rule so don't point them out to me. But it seems like getting to roll att rolls twice is much more statistically beneficial than getting to roll skill checks twice due to the idea that your skill bonuses are going to be the majority of the time higher than normal OR lower than normal, and your attack bonuses are usually going to require you roll a 8-13 or higher, making the middle ground more important. Am I using bad logic here?

and btw thanks chrono for the good ol fashioned number crunching!

[DracoSuave]

Originally Posted by shroom1000

It does not have any logic for natural 1s or natural 20s, just the chance of hitting.

The chance of getting a natural 20 is 9.75% (see, you calculated that.)

The chance of getting a natural 1 is exactly 1/400, because the only way to do so is by rolling two 1s.

Yes, Avengers hit -a lot.-

[FrozenChrono]

as you can see with firelance's conversion to static bonus's it's true that at the high and low end a +2 or +3 can be better. I guess it would depend on how trained you are with the skill and how your DM sets skill DC's.

I set mine with a mindset that

easy = 6 or 7+1/2 level (someone untrained should be able to do it)
medium = 11 or 12 +1/2 level (someone trained or with a high stat and can do it)
hard = 17 or 18 +1/2 level +2 per tier (someone trained with a high stat can do it)
very hard = 23 to 25 + level -2 per tier (someone trained with a main stat, an item bonus, specialization, a racial and/or background bonus. Basically a near maxed out skill. This is for extra special results in challenges or to do really cool difficult things)

I know it's not by the book, but it gives each of these character types about a 2/3-1/2 chance of success at each level and scales way better than the given DC's

[Flipguarder]

ToHit improvement from reroll
1 0
2 4.75%
3 9.00%
4 12.75%
5 16.0%
6 18.75%
7 21.00%
8 22.75%
9 24.00%
10 24.75%
11 25.00%
12 24.75%
13 24.00%
14 22.75%
15 21.00%
16 18.75%
17 16.00%
18 12.75%
19 9.00%
20 4.75%

So say I need a 13 to hit. Having 2 d20 rolls on that specific attack will net me an overall bonus of 24% or about +5 to hit (24/10 X 2).
I think i got that right.