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The Great Weapon Fighting Fighting style states the following:

When you roll a 1 or 2 on a damage die for an attack you make with a melee weapon that you are wielding with two hands, you can reroll the die and must use the new roll. The weapon must have a two-handed or versatile property for you to gain this benefit.

How much does this ability increase the average damage of its wielder?

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3 Answers 3

up vote 16 down vote accepted

I've forgotten the formal proof for this, but hopefully this is correct:

Consider a D6 (for the sake of concrete language).

When you roll a 1, you reroll the die and keep the result. This produces an average value of 3.5, and happens 1/6 of the time.

When you roll a 2, you reroll the die and keep the result (even if it's lower). This produces an average value of 3.5, and happens 1/6 of the time.

When you roll a 3, you keep the result. This produces an average value of 3, and happens 1/6 of the time.

And so on.

This gives the following formula for the average of the D6:

(3.5 + 3.5 + 3 + 4 + 5 + 6) / 6 = 4.1̅6

Working similar formulas for the other dice, we get this table:

Die   Avg.    Δ
---   ----   ----
d4    3.00   0.50
d6    4.1̅6   0.6̅6
d8    5.25   0.75
d10   6.30   0.80
d12   7.3̅3   0.8̅3

Dice are independent. 2D6 will have an average value of 2 * 4.1̅6, or 8.3̅3.

Common weapon average damage (Great Weapon Fighting):

  • Greatsword (2d6): 8.3̅3 (Δ1.3̅3)

  • Greataxe (1d12): 7.3̅3 (Δ0.8̅3)

  • Longsword (1d10): 6.30 (Δ0.80)

  • Smite (level 1, 2d8): 10.50 (+ weapon damage) (Δ1.5)

Observations:

  • The ability works out to about a +1 to damage.

  • It scales to almost a +3 when smiting. The more dice you add (high level smite, for example), the better the ability.

  • The bonus is "swingy." It can range from a -2 to a +10 on 2D6, for example.

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AceCalhoon's answer has the numbers, but I think it is useful to illustrate it with probability graphs. Here is how the probability distributions of two common damage dice, d12 and 2d6, change.

d12 distribution D12: The results 1 and 2 simply become very unlikely, boosting the probability of the rest.

2d6 distribution 2D6: Here the effect doesn't look linear. 2-5 all become much less likely. The peak is skewed from 6-8 to roughly 7-10.

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Using the same math as AceCalhoon, here is your relative damage increases in percentage.

d10: +14.5% damage

d8: +16.6% damage

d6: +19% damage

d4: +20% damage

However this only applies to damage dice, not flat damage bonuses. How many damage dice or flat bonuses you have depends greatly on your build (Great Weapon Master's +10 damage has no synergy, but Paladin's Smite and Crusader's Mantle do).

In general, if you use a greatsword, you can expect around a 10% to 15% overall damage boost from this fighting style.

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