calculating damage with Great Weapon Fighting style

[eldamir]

I'll admit it, I'm a bit of a numbers geek (but it's been waaaay too long since college and actually using math to be a math geek).
So, I was playing around with a spreadsheet to calculate damage and can't figure out how to account for rerolling 1's and 2's (I know that, statistically, it's 33.33% of the time that it'll happen)
Say, for instance, hitting with a greatsword at 20 str, you'd have:
2d6+5 for damage, which would be a minimum of 7 and max of 17 without accounting for rerolling a 1 or 2.
min: (2*1)+5 = 7
max: (2*6)+5 = 17
So, if 33% of the time, you're rolling an additional 1d6 for each of those dice and throwing out the original, how would that work?

[Rysto]

Let's look at re-rolling d6s. 2/3 times, you roll 3+. The average of those rolls is 4.5. You can come up with this average value in a number of ways, but in the end it winds up being the average of the die (3.5) +1 .

1/3 times, you roll a 1 or 2 and reroll the d6. The average roll of a d6 is 3.5. So 1/3 of the time the average roll is 3.5.

Now you just have to multiply the probability of each roll by the average:

2/3 * 4.5 + 1/3 * 3.5 = 4.166...

So on average you deal 4.166 per d6 (8.33... for a 2d6 weapon).


For a greataxe (d12), it's:
5/6 * 7.5 + 1/6 * 6.5 = 7.3333...

[Corran]

You are looking to find what is the expected value (pardon if the term is not the correct one, I am not 100% sure of the englisg terminology). To do that, you need to multiply each possible value its probability (ie the probability that this particular value is the result), and then add everything together.

Here is an example (no rerolling) for a d6.
The possible values are {1,2,3,4,5,6}. Each value has probability 1/6 (unbiased die) to show up as the result of rolling the die. So to find the expected value (average), you do as suggested above, meaning that average= (1/6)*1 + (1/6)*2 + ... + (1/6)*6 = 3.5.

Now, rerolling 1's and 2's only once (meaning that if for example we reroll a 2 and get a 1, we have to keep the 1), that changes the probabilities a bit.
Now to get a 1 as the resut, we need to roll 1 or 2 on our first attempt, and then we need to roll 1 on the second roll. And since there exsists independence, we multiplay to get the probability of 1 being the result, which is (2/6)*(1/6) = 2/36. For the same reason, this is also the probability of getting a 2.
What is the probability of getting a 3 as the damage output? We can either roll a 3 (that has probability 1/6), or we can roll a 1 or 2, and then roll 3 on the second roll, which means a probability of 2/36 in this second case. So we add them to get the probability that 3 is the result, which is 1/6 + 2/36 = 8/36. With the same reasoning, we get that the probability of rolling a 4, 5 and 6 respectively, is also 8/36.
Do the addition to check that all probabilities add up to 1 (they do), so that you can be sure you did not make any mistakes.
Now to find the expected value of rolling a d6 with one-time rerolls on 1's and 2's, you multiply each value with its respective probability and then you add them. So that is, (2/36)*1 + (2/36)*2 + (8/36)*3 + ... + (8/36)*6 = 150/36 = 4.16

[bid]

Another simple way to evaluate this:
- average 1d6 = 3.5
- average rolling 1-2 is 1.5
- rolling 1-2 on 1d6 = 1/3 of the time
- GWF gain = 1/3 * (3.5 - 1.5) = 2/3

It's the same thing for every dice:
- 1d4 = 1/2
- 1d6 = 2/3
- 1d8 = 3/4
- 1d10 = 4/5
- 1d12 = 5/6

[Lollerabe]

Took me a sec to find it, but here : http://www.giantitp.com/forums/showthread.php?486153-5e-GWF-and-Divine-Smite

While it isn't what you are asking about, it shows many of the numbers you are looking for - or so I hope

Edit: sorry reread the thread and dosent show that much math, this might help though

http://rpg.stackexchange.com/questions/47172/how-much-damage-does-great-weapon-fighting-add-on-average

[eldamir]

Thanks for the input, folks.
I ended up with the following to calculate average damage (not including crits, hit chance, etc. - that's a whole different piece of the spreadsheet!):

A5 = number of dice (i.e. 2)
C5 = type of dice (i.e. 6 for d6)
E5 = modifier (str/dex mod + duelist / GWM)
V6 = 'yes' or 'no' for Great Weapon Fighting

2d6+5 then gives
$A5*(IF($V$6="yes",((($C5-2)/$C5)*(AVERAGE(3,$C5))+(AVERAGE((1-($C5-2)/$C5)*(AVERAGE(1,$C5))))),AVERAGE(1,$C5)))+$E5
2*(((6-2)/6)*(AVERAGE(3,6))+(AVERAGE((1-(6-2)/6)*(AVERAGE(1,6)))))+5
2*((4/6)*(average(3,6))+(average((1-4/6)*(average(1,6)))))+5
2*((4/6)*(4.5)+(1.1667)*(3.5))+5
13.333 avg damage

1) 20 STR + No GWM
http://i.imgur.com/RnGN6lL.jpg

2) 20 STR + GWM
http://i.imgur.com/yWsTal8.jpg

3) 20 STR + GWM + Surprise
http://i.imgur.com/LE8lVRS.jpg

[Foxhound438]

Others have shown the math, the adjusted values for common 7-set dice are as follows:

d4:3
d6:4.16667
d8:5.25
d10:6.3
d12:7.33333

Note that the increased average is due to an upshifting of possible damage rolls away from 1 or 2 and towards everything else. basically you have a much lower chance of having a dud hit.

[bid]

A5 = number of dice (i.e. 2)
C5 = type of dice (i.e. 6 for d6)
E5 = modifier (str/dex mod + duelist / GWM)
V6 = 'yes' or 'no' for Great Weapon Fighting

$A5 * (
($C5 + 1)/2
+ IF($V6="yes", ($C5-2)/$C5, 0)
) + $E5

[R.Shackleford]

Hey kryx where the hell are you?

[eldamir]

$A5 * (
($C5 + 1)/2
+ IF($V6="yes", ($C5-2)/$C5, 0)
) + $E5


that's quite a bit simpler than the mess of an equation that I came up with!

[Kryx]

Hey kryx where the hell are you?
On vacation, but it's a very simple formula. "AVERAGE(3.5,3.5,3,4,5,6)" for example.

Or the more complicated spreadsheet one provided above.

[R.Shackleford]

On vacation, but it's a very simple formula. "AVERAGE(3.5,3.5,3,4,5,6)" for example.

Or the more complicated spreadsheet one provided above.

Who said you could take a vacation?