Here's the math on Chromatic Orb. It uses a known math problem called the birthday problem used to calculate the probability that at least 2 from a set of a certain size will be the same.
The probability that at least 2 out of (n)d8 will be the same number:
And then on top of that, at most levels there is roughly an 88% chance to hit with advantage, which a sorcerer would have using Innate Sorcery. Or 65% chance without, if you're a Wizard instead.
Rollback Post to RevisionRollBack
How I'm posting based on text formatting: Mod Hat On - Mod Hat Off
Yeah, I wanted to create actual comparisons between two spells, not just the probabilities, so people can compare exactly what the DPR differences are. I can break it down now and calculate Halfling Luck, Heroic Insp, etc.
Same but with Advantage. Chromatic orb single-target damage doesn't jump much, but it improved exponentially where multiple targets are available. With Innate Sorcery, this spell is a must-have crowd-control spell.
1d8 Firebolt
4.62 Average Damage per Cast
5.50 Average Damage per Target Hit
2d8 Firebolt
9.23 Average Damage per Cast
11.00 Average Damage per Target Hit
3d8 Firebolt
13.87 Average Damage per Cast
16.50 Average Damage per Target Hit
4d8 Firebolt
18.47 Average Damage per Cast
22.00 Average Damage per Target Hit
Level 1 Chromatic Orb:
17.03 Average Damage per Cast
15.07 Average Damage per Target Hit
Level 2 Chromatic Orb:
30.75 Average Damage per Cast
20.09 Average Damage per Target Hit
Level 3 Chromatic Orb:
52.51 Average Damage per Cast
25.11 Average Damage per Target Hit
Level 4 Chromatic Orb:
82.52 Average Damage per Cast
30.15 Average Damage per Target Hit
Level 5 Chromatic Orb:
115.58 Average Damage per Cast
35.15 Average Damage per Target Hit
Level 6 Chromatic Orb:
148.11 Average Damage per Cast
40.18 Average Damage per Target Hit
Level 7 Chromatic Orb:
178.20 Average Damage per Cast
45.20 Average Damage per Target Hit
Level 8 Chromatic Orb:
208.85 Average Damage per Cast
50.23 Average Damage per Target Hit
Level 9 Chromatic Orb:
239.75 Average Damage per Cast
55.25 Average Damage per Target Hit
I am very interested on how you found this formula!
Can you tell me where you got it or how you found it?
It's the birthday problem. It's based on 2+ people out of a group of size n sharing a birthday, but it works for 2+ dice out of a pool of size n sharing a value too.
Kind of late here, but I'm seeing a problem. I may be wrong, but here's what I see.
We assume a 65% chance to hit, and for simplicity I'm not factoring in crits.
So average damage for a single orb at 1st level starts at 13.5 (average of 4d8) * 0.65 = 8.775
For average damage to targets after the first, you not only have to multiply that 13.5 by the chance to jump (based on level), but you have to multiply by the chance to hit for each target up until you are at. In other words, your average damage to hit a second target with a 1st level spell is 13.5 * 0.344 (chance to jump) * 0.65^2 (chance to hit the first target, then chance to hit this target). A jump to a third target is * 0.65^3, etc.
This is because it only jumps if you hit the target before it. If you didn't have to hit each target in turn, then you could just use * 0.65 (and the chance to jump) for each target.
Kind of late here, but I'm seeing a problem. I may be wrong, but here's what I see.
We assume a 65% chance to hit, and for simplicity I'm not factoring in crits.
So average damage for a single orb at 1st level starts at 13.5 (average of 4d8) * 0.65 = 8.775
For average damage to targets after the first, you not only have to multiply that 13.5 by the chance to jump (based on level), but you have to multiply by the chance to hit for each target up until you are at. In other words, your average damage to hit a second target with a 1st level spell is 13.5 * 0.344 (chance to jump) * 0.65^2 (chance to hit the first target, then chance to hit this target). A jump to a third target is * 0.65^3, etc.
This is because it only jumps if you hit the target before it. If you didn't have to hit each target in turn, then you could just use * 0.65 (and the chance to jump) for each target.
Did I make a mistake?
I'm not sure what "problem" you're seeing? I ran the numbers with those probabilities (and crits) and got pretty close to what the OP has (also, it's 3d8 on the level 1 cast, but the 13.5 is correct anyway):
Average damage per cast = 11.14 (compared to OP's 10.86) Average damage per hit = 14.54 (compared to OP's 14.64)
It appears all of that has already been factored in.
MATH (please check me!):
Damage per cast = (13.5 * 0.60) + (27 * 0.05) + (13.5 * 0.344 * 0.60^2) + (27 * 0.344 * 0.05^2) = 11.14 13.5 average damage (27 for crit), 34.4% chance to bounce, 60% chance to hit on each hit, 5% chance to crit on each hit
Damage per hit = (13.5 * 0.923) + (27 * 0.077) = 14.54 13.5 average damage (27 for crit), 92.3% to hit normally, 7.7% chance to crit on hit
Okay, I see. I was looking at the post that had damage per cast with advantage, and missed that that post was including advantage (which is where the spell shifts into ludicrous speed).
Rollback Post to RevisionRollBack
To post a comment, please login or register a new account.
As per: Stats
Using AC of 20, and Attack Bonus of 11 (Crit range is 20), no advantage, assuming enough targets exist for it to jump.
1,000,000 iterations, so it should be very accurate
Firebolt for Comparison:
1d8 Firebolt
2d8 Firebolt
3d8 Firebolt
4d8 Firebolt
Level 1 Chromatic Orb:
Level 2 Chromatic Orb:
Level 3 Chromatic Orb:
Level 4 Chromatic Orb:
Level 5 Chromatic Orb:
Level 6 Chromatic Orb:
Level 7 Chromatic Orb:
Level 8 Chromatic Orb:
Level 9 Chromatic Orb:
-
View User Profile
-
View Posts
-
Send Message
ModeratorHere's the math on Chromatic Orb. It uses a known math problem called the birthday problem used to calculate the probability that at least 2 from a set of a certain size will be the same.
The probability that at least 2 out of (n)d8 will be the same number:
Chromatic Orb
Level 1, 2 targets, 3d8, 1-(8!/(8-3)!)/(8^3) = 34.4% chance to bounce
Level 2, 3 targets, 4d8, 1-(8!/(8-4)!)/(8^4) = 59%
Level 3, 4 targets, 5d8, 1-(8!/(8-5)!)/(8^5) = 79.5%
Level 4, 5 targets, 6d8, 1-(8!/(8-6)!)/(8^6) = 92.3%
Level 5, 6 targets, 7d8, 1-(8!/(8-7)!)/(8^7) = 98.1%
Level 6, 7 targets, 8d8, 1-(8!/(8-8)!)/(8^8) = 99.8%
Level 7, 8 targets, 9d8, at 100%
Level 8, 9 targets, 10d8, at 100%
Level 9, 10 targets, 11d8, at 100%
And then on top of that, at most levels there is roughly an 88% chance to hit with advantage, which a sorcerer would have using Innate Sorcery. Or 65% chance without, if you're a Wizard instead.
Homebrew Rules || Homebrew FAQ || Snippet Codes || Tooltips
DDB Guides & FAQs, Class Guides, Character Builds, Game Guides, Useful Websites, and WOTC Resources
Yeah, I wanted to create actual comparisons between two spells, not just the probabilities, so people can compare exactly what the DPR differences are. I can break it down now and calculate Halfling Luck, Heroic Insp, etc.
Same but with Advantage. Chromatic orb single-target damage doesn't jump much, but it improved exponentially where multiple targets are available. With Innate Sorcery, this spell is a must-have crowd-control spell.
1d8 Firebolt
2d8 Firebolt
3d8 Firebolt
4d8 Firebolt
Level 1 Chromatic Orb:
Level 2 Chromatic Orb:
Level 3 Chromatic Orb:
Level 4 Chromatic Orb:
Level 5 Chromatic Orb:
Level 6 Chromatic Orb:
Level 7 Chromatic Orb:
Level 8 Chromatic Orb:
Level 9 Chromatic Orb:
I am very interested on how you found this formula!
Can you tell me where you got it or how you found it?
-
View User Profile
-
View Posts
-
Send Message
ModeratorIt's the birthday problem. It's based on 2+ people out of a group of size n sharing a birthday, but it works for 2+ dice out of a pool of size n sharing a value too.
https://en.wikipedia.org/wiki/Birthday_problem
or
https://mathworld.wolfram.com/BirthdayProblem.html
Homebrew Rules || Homebrew FAQ || Snippet Codes || Tooltips
DDB Guides & FAQs, Class Guides, Character Builds, Game Guides, Useful Websites, and WOTC Resources
Thank You! I will definitely look into this!
Kind of late here, but I'm seeing a problem. I may be wrong, but here's what I see.
We assume a 65% chance to hit, and for simplicity I'm not factoring in crits.
So average damage for a single orb at 1st level starts at 13.5 (average of 4d8) * 0.65 = 8.775
For average damage to targets after the first, you not only have to multiply that 13.5 by the chance to jump (based on level), but you have to multiply by the chance to hit for each target up until you are at. In other words, your average damage to hit a second target with a 1st level spell is 13.5 * 0.344 (chance to jump) * 0.65^2 (chance to hit the first target, then chance to hit this target). A jump to a third target is * 0.65^3, etc.
This is because it only jumps if you hit the target before it. If you didn't have to hit each target in turn, then you could just use * 0.65 (and the chance to jump) for each target.
Did I make a mistake?
I'm not sure what "problem" you're seeing? I ran the numbers with those probabilities (and crits) and got pretty close to what the OP has (also, it's 3d8 on the level 1 cast, but the 13.5 is correct anyway):
Average damage per cast = 11.14 (compared to OP's 10.86)
Average damage per hit = 14.54 (compared to OP's 14.64)
It appears all of that has already been factored in.
MATH (please check me!):
Damage per cast = (13.5 * 0.60) + (27 * 0.05) + (13.5 * 0.344 * 0.60^2) + (27 * 0.344 * 0.05^2) = 11.14
13.5 average damage (27 for crit), 34.4% chance to bounce, 60% chance to hit on each hit, 5% chance to crit on each hit
Damage per hit = (13.5 * 0.923) + (27 * 0.077) = 14.54
13.5 average damage (27 for crit), 92.3% to hit normally, 7.7% chance to crit on hit
Okay, I see. I was looking at the post that had damage per cast with advantage, and missed that that post was including advantage (which is where the spell shifts into ludicrous speed).