How to include successful saves when calculating Fireball's average damage?

Question

I want to know how to calculate the average damage of a spell that also deals half damage on a successful save. For this example, I'll be taking the most popular evocation spell, Fireball.

As far as calculating damage goes, I know how to calculate the expected damage of attacks using an attack roll, using the following formula:

Expected damage = Probability x Damage + Crit chance x Additional damage on crits
Probability = (21 - target's AC + attacker's attack roll modifiers) x 5%

Now I would assume that you just need to reverse the probability formula to calculate a spell that forces a saving throw's chance of success, like so:

Probability = 1 - (21 - your save DC + target's save modifiers) x 5%

However, calculating Probability x Damage (omitting the crit chance in the process) using the above formula does not take into account the half damage dealt on a successful save. So how to take this into account when calculating expected damage of spells like Fireball?

Answer 1

You need to consider both cases

The half damage on a successful save can be included in the calculation just like crit damage.

You know that the probability ("p") of successful + unsuccessful is 100 %.

p(successful) + p(unsuccessful) = 100 %

Therefore (100 %-p(unsuccessful)) is the successful probability. You can multiply this probability with half damage to include the respective contribution. Contrary to the crit case where crit is a subset of hits, successful and unsuccessful are independent and you need to use the full amount of (the half) damage rather than the difference.

damage(expected) = p(successful) x half damage + p(unsuccessful) x full damage

Answer 2

Step 1: Failure chance.

First, we need to know the probability of a successful save. This formula uses DC, which is the spell save DC, and MOD, which is whatever modifier gets added to the target creature’s saving throw:

$$P(s)=\frac{21-DC+MOD}{20}$$

So our probability of failure is then:

$$P(f)=1-\frac{21-DC+MOD}{20}$$

Step 2: Expected failures on N identical targets.

Here we assume that our fireball targets all have the same MOD added to their saving throw. The expected number of failures for a binomial distribution is:

$$E(f)=N\cdot P(f)$$

So naturally, the expected number of successes is:

$$E(s)=N-N\cdot P(f)=N\cdot P(s)$$

Step 3: Expected damage.

Assume now that we roll D damage. We just multiply D by the number of failures, and half of D by the number of successes, and add those two together. Then the expected damage of our fireball on N identical targets is:

$$D\cdot (E(f)+0.5E(s))=D\cdot E(f)+0.5D\cdot E(s)$$