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:
- Increase the static damage for the attack (as in your firebolt vs dagger example) — Larger static variables weight it towards average damage.
- Reduce the damage die (as in your firebolt vs dagger example) — Smaller dice have a lower variance.
- 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
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.