Background Theory

Broadly when choosing to optimise damage in a single round there are five variables you need to take account of:

• Average damage per round (I’m assuming that chance to hit has been rolled up into this variable)
• Damage variance
• Damage immunities/vulnerabilities
• Enemy HP
• Have you Crit’d (This fact significantly increases the math required.)

When you optimise DPR, you are optimising only one/two of these five potential variables (Criting can alter the ADPR).

If you are only rolling one dice, your damage variance on that attack will be relatively high.

To reduce the variance of the attack there are three broad strategies:

1. Increase the static damage for the attack (as in your firebolt vs dagger example) — Larger static variables weight it towards average damage.
2. Reduce the damage die (as in your firebolt vs dagger example) — Smaller dice have a lower variance.
3. Increase the number of dice you use in the attack (the more dice you roll, the more likely you are to get the average damage for a given attack) - This is an application of the law of large numbers.

In the situation where there is a damage type resistance/vulnerability then the choice between the two becomes clearer as you will be halving/doubling the result from the damage dice which will decrease/increase your DPR.

To answer whether or not you are more likely to beat a specific enemy depends on the HP, and variability in that enemies HP (determined by their hit dice). The same two considerations apply here.

If you are fighting an enemy with a large amount of hit dice, then by the same application of the law of large numbers you are more likely to be fighting an average specemin of the enemy.

Similarly if you are fighting an enemy with a large fixed HP component then you are similarly more likely fighting an average enemy.

Application to your example

The Firebolt has a range of damage values when it hits of 1-10 (with a 10% chance of each)

The dagger has a different range of damage values when it hits 4-7 (with a 25% chance of each).

A Goblin’s HP is calculated by rolling 2d6 (the 7 HP in the Monster Manual is simply the average value of this distribution). This gives the Goblin Monster a range of Hit Point values between 2 & 12 HP (obviously the extremes of 2 and 12 are pretty unlikely).

Both attacks have a 45% chance of a non-critical hit and a 5% chance of a critical hit.

The rest of this analysis will assume that we have hit (that makes the math slightly earier, and we can convert it into number of rounds by using this information later).

As a result, when we hit that translates to a 90% of hits are a non-crit, and 10% are a crit.

I’m also going to assume, for simplicities sake, that we are doubling the result of the dice when we crit, instead of doubling the number of dice we roll.

Vanilla Attacks

1 Hit to Kill

Taking your specific example of a Goblin (7 HP). When our attacks hit

• The Firebolt has a 40% chance of killing the Goblin in 1 non-critical hit (7–10)
• The Firebolt also has a 70% chance of killing the Goblin in 1 critical hit (4–10) x 2
• The dagger has a 25% chance of killing the Goblin in 1 non-critical hit (4 + 3)
• The dagger has a 75% chance of killing the Goblin in 1 critical hit ( 2 x (2–4) + 3)

Taking all of this into account, the 1 hit kill percentages are:

• Firebolt: 40% * 90% + 70% * 10% = 43%
• Dagger: 25% * 90% + 75% * 10% = 30%

3 Hits or more to Kill

At the other extreme, the chance of it taking more than two hits to fell the Goblin.

• Dagger: 0% (Even if we roll the minimum non-crit damage two hits will kill it.)
• Firebolt: We have to roll a total of 6 or less between the first two hits. With crits this is a more complicated piece of maths.

Ways we can get this total with Firebolt

2 x Crits:

• 1 x 2 + 1 x 2: ( 10% * 10% * 10% * 10% )
• 1 x 2 + 2 x 2: ( 10% * 10% * 10% * 10% )
• 2 x 2 + 1 x 2: ( 10% * 10% * 10% * 10% )

1 x Crit:

• 1 x 2 + 1: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 2: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 3: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 4: ( 10% * 10% * 10% * 90% )
• 1 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 2 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 3 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 4 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 2 x 2 + 1: ( 10% * 10% * 10% * 90% )
• 2 x 2 + 2: ( 10% * 10% * 10% * 90% )
• 1 + 2 x 2: ( 10% * 90% * 10% * 10% )
• 2 + 2 x 2: ( 10% * 90% * 10% * 10% )

0 x Crits:

• 1 + 1: ( 10% * 90% * 10% * 90% )
• 1 + 2: ( 10% * 90% * 10% * 90% )
• 1 + 3: ( 10% * 90% * 10% * 90% )
• 1 + 4: ( 10% * 90% * 10% * 90% )
• 1 + 5: ( 10% * 90% * 10% * 90% )
• 2 + 1: ( 10% * 90% * 10% * 90% )
• 2 + 2: ( 10% * 90% * 10% * 90% )
• 2 + 3: ( 10% * 90% * 10% * 90% )
• 2 + 4: ( 10% * 90% * 10% * 90% )
• 3 + 1: ( 10% * 90% * 10% * 90% )
• 3 + 2: ( 10% * 90% * 10% * 90% )
• 3 + 3: ( 10% * 90% * 10% * 90% )
• 4 + 1: ( 10% * 90% * 10% * 90% )
• 4 + 2: ( 10% * 90% * 10% * 90% )
• 5 + 1: ( 10% * 90% * 10% * 90% )

All of that is:

• 3 * ( 10% * 10% * 10% * 10% ) + 12 * ( 10% * 10% * 10% * 90% ) + 15 * ( 10% * 90% * 10% * 90% )

Which totals to 13.26%

Final Totals

As a result the probabilities of killing the Goblin with a repeated attack type are:

1 hit to kill:

• Firebolt: 43%
• Dagger: 30%

2 hits or less to kill:

• Firebolt: 86.74%
• Dagger: 100%

3+ hits or more to kill:

• Firebolt: 13.26%
• Dagger: 0%

Mix & Match

This is of course assuming you don’t mix and match attacks.

If we allow mixing and matching your choices change. For example, if you hit with the Firebolt or the Dagger for 6 damage, it doesn’t matter which attack you hit with next.

Similarly if you hit with the firebolt for 3 or more (80% probability for the hit), using the dagger next hit guarantees the kill.

If you hit with the firebolt for 1 on the first hit (10%) then your probabilities are:

• Firebolt: 50% (6 - 10) * 90% + 80% (6 - 20) * 10% = 53%
• Dagger: 50% (6 - 7) * 90% + 75% (7 - 11) * 10% = 52.5%

The Dagger’s smaller variance will probably give it enough to pip the Firebolt (but it’s close).

If you hit with the firebolt for 2 damage on the first hit (10%*90% + 10%*10% = 10%) then your probabilities are:

• Firebolt: 60% (5 - 10) * 90% + 80% (6 - 20) * 10% = 62%
• Dagger: 75% (5 - 7) * 90% + 100% (4 - 11) * 10% = 77.5%

You can see where this is trending…

Consolidated Percentages

1 Hit to Kill

• Firebolt: 43%
• Dagger: 30%

2 Hits or less to Kill

• Firebolt, then Dagger: (80%*90% + 90%*10%) * 100% + 10% * 52.5% + 10% * 77.5% = 94%
• 2 x Firebolt: 86.74%
• 2 x Dagger: 100%
• Dagger then Firebolt (we don't care as doing this would be stupid)

3 Hits or less to Kill

• Firebolt, Dagger, Dagger: 100%

3 Hits or more to Kill

• 3 x Firebolt: 13.26%

Strategy against a 7 HP Goblin

Hit with Firebolt first, as it gives you a better chance of one-shot killing the Goblin.

With two hits the probabilities of killing the Goblin are: Firebolt + Dagger = 94% Dagger + Dagger = 100%

If you hit with the Firebolt on your first hit, and it doesn't kill the Goblin (and the Goblin has 7HP) then you are better off switching your second attack to be the dagger, as you have a higher (and less variable) probability of killing the Goblin with the Dagger on the second hit.

Given that the increase of kill % with 1 hit on a Firebolt (13%) is greater than the decrease (6%) in % we get over two rounds from not doing Dagger, Dagger, Optimal play is to do Firebolt + Dagger.

On average how many rounds does it take for you to hit the Goblin?

We effectively have repeated trials to success of a iid Bernoulli random variables with p=0.5. The expected waiting time until a success (a hit) for this type of process is given by the expected value of a Geometric distribution, with p=0.5.

E[Rounds to Hit] = 1/0.5 = 2 rounds.

Thus the expected rounds to get 2 hits is 4.

Conclusion

This sort of spread will change depending on the actual HP of the Goblin.

• Lower than 7 HP will lean towards you using the Dagger for both attacks (6HP is the point where the one shot probabilities for both Firebolt and Dagger are close enough (53% vs 52.5%) that the consistency of the Dagger leans in its favour).
• Higher than 7HP Firebolt first (and potentially 2nd) Dagger later will be the better combo.

1. Treat healing/round as damage/round for calculations

While the DMG doesn't give a clear answer, Spellcasting's effect on CR (DMG p.281) states:

\$\begin{array}{|l|l|l|} \hline \textbf{Name} & \textbf{Example Monster} & \textbf{Effect on Challenge Rating }\\ \hline \text{Spellcasting} & \text{Lich} & \text{See step 13 under "Creating a Monster Stat Block"}\\ \hline \end{array} \$

Where Step 13 states:

Innate Spellcasting and Spellcasting. The impact that the Innate Spellcasting and Spellcasting special traits have on a monster's challenge rating depends on the spells that the monster can cast Spells that deal more damage than the monster's normal attack routine and spells that increase the monster's AC or hit points need to be accounted for when determining the monster's final challenge rating.

This tells us that spells in general should influence CR (Unlike some other monster features), it's just lacking by how much.

We can however reverse engineer a monster in the book to see how its CR is affected by spells such as these, by determining its CR before taking into account healing and other beneficial magic, and comparing it to its actual CR. Let's take this Couatl into consideration. Most of its spells are healing and damage mitigation, and it gets a large amount of them, leaving it as a good starting point. Breaking it down based off the Monster Stats by CR table (DMG p.274), we have:

Defensive CR: With 97 HP* the Couatl lands it in the CR 2 category. Its AC of 19 is 6 higher than the expected CR, leaving it Defensively CR 5

Offensive CR: At 10 damage/round, Its offensive CR starts as a 1. Its attack bonus of +6 is 4 higher than the suggested, leaving it Offensively CR 3

Averaging these out, the Coautl comes to CR 4. This is without factoring in any amount of healing or beneficial magic cast.

The next step is to see how far we can deviate from this, to see if magical healing increases CR on top of normal calculations. The most apparent starting point is determining if we can raise it's effective HP based off of healing magic, and still stay within the confines of it being a CR 4 creature. Looking at the HP category it fell under, we see a range of 86-100 HP. Any more and we have the potential to bump up it's CR. Seeing that the Couatl is already at 97 HP, this means an increase of any more than 3 effective HP will cause the Defensive CR to increase to 6, increasing the Couatl's total CR to 5.

Since this is not the case, it can be inferred that healing magic does not affect the Defensive CR of a monster.

Now we see an example where a creature's CR is affected by its spellcasting. Take the Acolyte. With a defensive CR of 0 and being 27 HP below the threshold for increasing (Eliminating again the possibility of healing affecting Defensive CR, as using every spell slot to heal only achieves 24 points), the Acolyte needs an Offensive CR of 1/2 to meet its average CR of 1/4. This means 6-8 damage/round. Given it does 2 with a melee attack, 4 with it's only damaging spell, and heals 6 HP with a Cure Wounds, the Cure Wounds is the only option to put the offensive CR in line.

No examples given in the Monster Manual give creatures with the Aid spell, but due to it being a spell that's cast before combat begins, it stands to reason to work like Mage Armor (DMG p.276), in this event granting HP to go towards Defensive CR calculations (As there's next to no difference in a creature with 120 HP with Aid than there is with a creature having 125 HP without Aid). Whether it affects more than one creature (Potentially raising the effective HP boost to 10-15) is detailed below.

*Doubling it's effective HP, as the book suggests you should do for a CR 4 creature with immunity to nonmagical bludgeoning, piercing, and slashing damage among other immunities and resistances (DMG p.277), brings its Defensive CR to 10 after calculations, leaving the Couatl CR 7 by these notes. Why this step was omitted I don't know, but is beyond the scope of the question.

2. There is no clear answer

The only two Monster Features (DMG p.280) that revolve around area effects, the Breath Weapon and Death Burst, operate off the assumption that two creatures will be affected, so this is the best baseline to go by. It's unknown whether the caster should be included in the two creatures, or it should be the caster plus two creatures (Since neither given feature is sufficient to determine if the intent was to affect two creatures or affect two other creatures), so some discretion is needed.