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Assume the following character build:

Level 2 Variant Human Wild Magic Sorcerer with Magic Initiate Feat; relevant spells are Sacred Flame and Fire Bolt. While I don't think character level is relevant, it is included in case an ability plays a larger roll than anticipated.

Ability Scores:
STR: 8 (-1)
DEX: 14 (+2)
CON: 14 (+2)
INT: 10 (+0)
WIS: 12 (+1)
CHA: 14 (+2)

Spell Save DC: 8 + 2 + 2 = 12
Spell Attack Modifier: 2 + 2 = 4

Spell Save DC for Sacred Flame: 8 + 2 + 1 = 11

Magic Initiate (5e PHB page 168):

Choose a class{cleric}: You learn two cantrips of your choice from that class’s spell list. ... Your spellcasting ability for these spells depends on the class you chose: ... Wisdom for cleric

Sacred Flame (5e PHB page 272):

Cantrip.
Casting Time: 1 action.
Flame-like radiance descends on a creature that you can see within range. The target must succeed on a Dexterity saving throw or take 1d8 radiant damage. The target gains no benefit from cover for this saving throw.

Firebolt (5e PHB page 242):

Cantrip.
Casting Time: 1 action
You hurl a mote of fire at a creature or object within range. Make a ranged spell attack against the target. On a hit, the target takes 1d10 fire damage. A flammable object hit by this spell ignites if it isn't being worn or carried.

Shadow (53 MM page 269),

Armor Class 12
Hit Points 16 (3d8 + 3)
DEX: 14 (+2)
Damage Vulnerabilities radiant
Damage Resistances ... fire ...

Damage Resistance and Vulnerability (53 PHB page 197)

If a creature or an object has resistance to a damage type, damage of that type is halved against it. If a creature or an object has vulnerability to a damage type, damage of that type is doubled against it.

Assuming that we don't have to worry about the Sorcerer taking any damage (just for the sake of the question, otherwise it boils down to the Sorcerer dying after a few rounds), which spell would be the better choice to deal the maximum amount of damage on average within an arbitrary but finite amount of rounds?

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    \$\begingroup\$ I suggest that you list the DC for Sacred Flame, or break it out, since you are casting it with Wisdom per the rule you cited for Magic Initiate. With a 14 Cha and a 12 Wis, your DC for the Dex save for Sacred Flame is one less than the DC for Firebolt. 8 + 2 + 1. Was that a deliberate omission or an oversight? \$\endgroup\$ – KorvinStarmast Jun 21 '17 at 20:17
  • \$\begingroup\$ No worries, I messed up the DC, and someone caught it. We help each other. (And if you go cleric, you can also choose Guidance which is an awesome utility cantrip that can help the whole party do things: ability checks get a +1-4 for the cost of a cantrip). This comment brought to you by Cleric's Local 407734. \$\endgroup\$ – KorvinStarmast Jun 21 '17 at 20:59
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Ideally, Sacred Flame kills it (roughly) twice as fast

A simple formula answers this, with X being the number of rounds needed to kill the Shadow.
1d10 averages 5.5, halved is 2.75
1d8 averages 4.5, doubled is 9

For Firebolt, you hit on 8-20 (chance for crit), so .65 x (5.5/2)(crit bonus)x(XFB) >= 16
(Need to hit AC 12; roll 1d20 +2 Int +2 proficiency)
(14 / 13 = crit factor; (x+1)/x for damage with the chance of crit thrown in)

For Sacred Flame, you hit on 13-20 (and no crit) .4 x (4.5 x 2)(XSF) >= 16
With a DC of 11 (you only have +1 due to your 12 Wisdom)
(The Shadow makes a dex saving throw versus the spell)

Solve for XFB: Solve for XFB: 16/(.65 x 2.75 x 1.077) = 8.31 rounds
Solve for XSF: 16/(.4 x 9) = 4.44 rounds

As a practical matter, being able to kill it in ~4 rounds rather than ~8 rounds means that your Sorcerer with that meager 16 HP has a far better chance to survive the fight before having no offensive capability.

Actions
Strength Drain. Melee Weapon Attack: +4 to hit, reach 5 ft., one creature.
Hit: 9 (2d6 + 2) necrotic damage.

Two hits and your Sorcerer is down.

This is an encounter were the Cleric's "turn undead" would be a handy skill to have.

This whole problem also presumes that you stay more than five feet away from the Shadow. In the case of firebolt, since that is a ranged spell attack rather than a spell that calls for a save, if the Shadow closes to melee range your firebolt attacks at disadvantage.

(SRD p. 103; spell attacks) Most spells that require attack rolls involve ranged attacks. Remember that you have disadvantage on a ranged attack roll if you are within 5 feet of a hostile creature that can see you and that isn’t incapacitated.

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    \$\begingroup\$ His cleric DC is 11, actually, not 13. The 40% to deal damage is right, though. \$\endgroup\$ – Szega Jun 21 '17 at 20:56
  • \$\begingroup\$ Your expected damage for Firebolt is too low. Firebolt has a 65% chance to hit, not 45%. \$\endgroup\$ – Nick Brown Mar 23 '18 at 13:57
  • \$\begingroup\$ Yeah. +4 to the attack vs a 12 AC. That means an 8 hits and a 7 misses, so 65% hit and 35% miss. Hmmm. We solved the same math problem different ways, but still have different answers (closer than before though). So one of us is slightly wrong still. I'll double-check mine and yours when I have time. \$\endgroup\$ – Nick Brown Mar 23 '18 at 14:54
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    \$\begingroup\$ Sorry, but this is math, so it should be right. 1.05 can only be the correct crit bonus if you always hit (21/20). The idea is that the crit dies as much extra damage as a normal hit (since there are no bonuses in this case), so you have to have use (x+1)/x not (x+1)/20 which expresses how much the average damage goes up for hits once you figure that 1 of them is actually a crit. Your 2nd calculation is now correct (with some rounding), but there's no need for the first wrong calculation now. \$\endgroup\$ – Nick Brown Mar 25 '18 at 4:13
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    \$\begingroup\$ Thanks. I appreciate your answer, but I have a thing about making sure that the math of these things is as accurate as it can be so we don't mislead those who come after. \$\endgroup\$ – Nick Brown Mar 25 '18 at 17:17
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Sacred Flame is superior to Firebolt

(even with a lower casting stat)

Sacred Flame has a 3.6 dpr vs a Shadow

Firebolt has a 1.9 dpr vs a Shadow

I posted this answer to correct some math problems in the other answers (fixed in the accepted answer now), even though all answers reached the same conclusion.

Firebolt is a ranged spell attack

  • your attack bonus is +4 (charisma+proficiency; 2+2)
  • the target AC is 12
  • you hit on an 8 and critical hit on a 20 (65% chance to hit the Shadow)
  • your damage is 1d10/2 so the average is 2.75 for a normal hit or 5.5 for a critical hit
  • your expected damage per round is 7/20*0 + 12/20*2.75 + 1/20*5.5 = 1.925 dpr

Sacred Flame is a spell which forces a dex save

  • your dc is 11 (wisdom+proficiency+8; 1+2+8)
  • the target's dex save modifier is +2
  • the target saves on a 9 (40% chance for the Shadow to fail its save)
  • your damage is 1d8*2 so the average is 9
  • your expected damage per round is 12/20*0 + 8/20*9 = 3.6 dpr

It will take you 4.44 rounds of average damage to kill a 16 hp shadow with Sacred Flame. If the shadow fails its save and max damage is rolled, you can kill it in 1 attack.

It will take you 8.31 rounds of average damage to kill a 16 hp shadow with Firebolt. With all critical hits and max damage, you will still need 2 rounds to kill the shadow.

Firebolt has a higher success rate (65%) than the Sacred Flame (40%) in this scenario. Also, Firebolt can crit. It is only better in this case because the Firebolt damage is halved and the Sacred Flame damage is doubled. Against a similar foe without these resistances or vulnerabilities, Firebolt would have a 3.85 dpr and Sacred Flame would only have 1.8 dpr.

It just goes to show, that when a normally weaker attack has a situational x4 multiplier vs a normally stronger attack, it can sometimes be better.

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  • \$\begingroup\$ And why not a +1 for your persistence in improving my answer? Thanks for your efforts. :) \$\endgroup\$ – KorvinStarmast Mar 25 '18 at 15:02
  • \$\begingroup\$ There are 2 creatures vulnerable to radiant damage from hundreds. You should mention that spending an ASI to get a cantrip that is based on a dump stat is a horrible idea 99% of the time. \$\endgroup\$ – András Nov 27 '18 at 11:04
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In this case, the shadow would take extra damage from the sacred flame from its vulnerability (2x) and less (half) damage from the firebolt.

Taking averages:

Sacred flame: 1d8x 2 = 4.5 * 2 = 9 dmg

Firebolt: 1d10 /5 = 5.5 / 2 = 3 dmg

So the sacred flame will, on average deal more damage. The save DCs and the attack rolls are relatively similar...

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    \$\begingroup\$ Not quite. The firebolt (65%) is successful 50+% more than the sacred flame (40%) in this scenario. Also, the firebolt can crit. You need to work out the expected damage. Sacred Flame is still better than firebolt, but only by about 2x, not 3x. \$\endgroup\$ – Nick Brown Mar 23 '18 at 13:51

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