At first level, is the fighter Great Weapon Fighting Style better than Barbarian Rage regarding 1st level characters with 2-handed weapons? Like in damage output.
The better one is the one you want. Do you want an active trait with limited uses that gives you a flat bonus to damage, or do you want an always active trait that reduces the likelihood of rolling poor damage?
Objectively saying one is better than the other is impossible because you can't guarantee the results of your rolls or the various trait affecting events that will occur in the campaign. A 95% chance to hit, should only miss 5% of the time, but it could still miss 100% of the time.
However, my subjective opinion is: Fighter is the better choice when using a weapon with a damage roll that has multiple dice, while Barbarian is better for singular damage dice rolls.
Edit: in other words: Greatswords for Fighters, Greataxes for Barbarians.
Critical hits. The features of Barbarian statistically favor weapons with a large, single damage die. Brutal Critical adds singular weapon damage dice, not multiplicative total weapon damage dice.
Brutal Critical (1) adds 1d12 to a Greataxe critical, or 1d6 to a Greatsword critical.
Brutal Critical (3) adds 3d12 to a Greataxe critical, or 3d6 to a Greatsword critical.
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You don't know what fear is until you've witnessed a drunk bird divebombing you while carrying a screaming Kobold throwing fire anywhere and everywhere.
Critical hits. The features of Barbarian statistically favor weapons with a large, single damage die. Brutal Critical adds singular weapon damage dice, not multiplicative total weapon damage dice.
Brutal Critical (1) adds 1d12 to a Greataxe critical, or 1d6 to a Greatsword critical.
Brutal Critical (3) adds 3d12 to a Greataxe critical, or 3d6 to a Greatsword critical.
While it is true, a critical hit is rather rare and doesn't really impact until you get +2 critical hit dice. Expanding your critical hit range does help as does gaining advantage on the to-hit role. If you only have +2 critical hit damage you need advantage or improved critical to get some substantial benefit from wielding a Great-axe. From +3 critical dice onward, the Great axe is better on average all the time.
Without expanded crit range Greatswords are the most consistent weapons, either fighter or barbarian. Great Weapon Fighting style benefit 2d6 more because the chance to roll 1's and 2's is just that much higher and it nets 1,33 more damage on average for 2d6, while only about 0,83. Yet because the probability curve isn't centered anymore the median value is more relevant and the minimum and maximum expected results are. Yet one needs some basic comprehension of statistics to understand this, so I won't goo too deep into this. The median for 2d6 moves from 7 to 9 and for 1d12 this is rather pointless as all values have the same chance to come up, but if we take the average median, this moves from 6,5 to 7,5. So the difference there is +2.
In concrete barbarian levels this translates to:
for Half-Orc barbarians the tipping point is 13th level to prefer the Great Axe over the Great Sword.
for other races this is 17th level.
If you have advantage and/or improved critical the tipping point drop to level 9 and 13 respectively.
Let's not forget the rest of both packages. Your DPR is 0 if you're down and/or dead.
Fighters generally have higher starting armor classes due to heavy armor and get a spontaneous self-heal, allowing them to continue adventuring at (almost) full power after short rests.
Barbs get temporary resistance to the most common damage types and advantage on strength checks, and their base hit die is bigger. Unarmored defense also helps mitigate the party's cost of upgrading equipment. Rage recharges on long rests.
They both have their strengths and weaknesses. Talk with your DM about the expected adventuring days s/he plans to have for the party. If the day expects 1-2 big fights between long rests, barb feels more reliable; 6-7 with 2+ short rests, I'd go with fighter.
At first level, is the fighter Great Weapon Fighting Style better than Barbarian Rage regarding 1st level characters with 2-handed weapons? Like in damage output.
The better one is the one you want. Do you want an active trait with limited uses that gives you a flat bonus to damage, or do you want an always active trait that reduces the likelihood of rolling poor damage?
Objectively saying one is better than the other is impossible because you can't guarantee the results of your rolls or the various trait affecting events that will occur in the campaign. A 95% chance to hit, should only miss 5% of the time, but it could still miss 100% of the time.
However, my subjective opinion is: Fighter is the better choice when using a weapon with a damage roll that has multiple dice, while Barbarian is better for singular damage dice rolls.
Edit: in other words: Greatswords for Fighters, Greataxes for Barbarians.
How does singular die rolls benefit the Barby?
Critical hits. The features of Barbarian statistically favor weapons with a large, single damage die. Brutal Critical adds singular weapon damage dice, not multiplicative total weapon damage dice.
Brutal Critical (1) adds 1d12 to a Greataxe critical, or 1d6 to a Greatsword critical.
Brutal Critical (3) adds 3d12 to a Greataxe critical, or 3d6 to a Greatsword critical.
You don't know what fear is until you've witnessed a drunk bird divebombing you while carrying a screaming Kobold throwing fire anywhere and everywhere.
Interesting.
While it is true, a critical hit is rather rare and doesn't really impact until you get +2 critical hit dice. Expanding your critical hit range does help as does gaining advantage on the to-hit role. If you only have +2 critical hit damage you need advantage or improved critical to get some substantial benefit from wielding a Great-axe.
From +3 critical dice onward, the Great axe is better on average all the time.
Without expanded crit range Greatswords are the most consistent weapons, either fighter or barbarian. Great Weapon Fighting style benefit 2d6 more because the chance to roll 1's and 2's is just that much higher and it nets 1,33 more damage on average for 2d6, while only about 0,83.
Yet because the probability curve isn't centered anymore the median value is more relevant and the minimum and maximum expected results are. Yet one needs some basic comprehension of statistics to understand this, so I won't goo too deep into this. The median for 2d6 moves from 7 to 9 and for 1d12 this is rather pointless as all values have the same chance to come up, but if we take the average median, this moves from 6,5 to 7,5. So the difference there is +2.
In concrete barbarian levels this translates to:
If you have advantage and/or improved critical the tipping point drop to level 9 and 13 respectively.
Let's not forget the rest of both packages. Your DPR is 0 if you're down and/or dead.
Fighters generally have higher starting armor classes due to heavy armor and get a spontaneous self-heal, allowing them to continue adventuring at (almost) full power after short rests.
Barbs get temporary resistance to the most common damage types and advantage on strength checks, and their base hit die is bigger. Unarmored defense also helps mitigate the party's cost of upgrading equipment. Rage recharges on long rests.
They both have their strengths and weaknesses. Talk with your DM about the expected adventuring days s/he plans to have for the party. If the day expects 1-2 big fights between long rests, barb feels more reliable; 6-7 with 2+ short rests, I'd go with fighter.