How good is the N avg(D) formula for hit points?

Recall the monster hit point formula N avg(D), from a previous post. N is the number of hits a monster takes before it dies, and avg(D) is the average amount of damage a player does per hit.

One can check how well this formula works with an online dice calculator.

Let's examine a case where a player does D=d8+4, avg(d8+4)=8.5 damage every time a player hits a monster. For a monster lasting N=4 hits until it dies, the monster should have 4(8.5) = 34 hit points.

Using the online dice calculator to calculate the total damage after 4 hits, one can see that 34 hit points is at the peak of the distribution for 4d8 + 16, which is distribution of possible total damage from a player doing d8+4 damage per hit for 4 hits. The probability of the total damage being greater than or equal to 34 after 4 hits, is 54.2%.

Next one can examine the distribution of possible total damage after 5 hits, which is 5d8 + 20. From this, the probability of the total damage being greater than or equal to 34 after 5 hits, is 96.09%.

Doing the same calculation for 6 hits, we get:

prob(total damage >= 34) = 99.97%

In general, one can do enough of these calculations and convince one's self that after N hits of a player doing D damage to a monster having N avg(D) hit points, the distribution of the total damage after N hits will be such that:

prob(total damage >= N avg(D)) ~ 50% - 55%

After N+1 or N+2 hits, the distribution of the total damage will be such that:

prob(total damage >= N avg(D)) ~ 90% - 100%.


If one tries hard enough, one can find situations where this doesn't quite hold exactly, such as the case of N=3, D=d4+1, avg(D)=3.5. In this case after 3 hits:

prob(total damage >= 10) = 68.75%.

After 4 hits, prob(total damage >= 10) = 98.05%.

Another extreme case which doesn't quite hold exactly, is the case of N=21, D=d12+7, avg(D)=13.5. In this case after 21 hits:

prob(total damage >= 283) = 52.5%.

After 22 and 23 hits,

22 hits: prob(total damage >= 283) = 81.36%
23 hits: prob(total damage >= 283) = 95.46%

Hit points of 4E monsters (part 2).

Let's look at the hit points of low level 4E monsters again.

From a previous post, the hit points of some favorite generic low level monsters are:

Kobold: 24, 27, 36 (level 1-2)
Goblin: 25, 29, 31 (level 1-2)

Let's assume a level 1 player with an at-will power using a weapon which does d8 damage, and the player's primary combat stat has a mod of +4. Hence the damage the player does with each hit is d8+4. The minimum damage per hit is 5 and the maximum damage per hit is 12. The low and upper limits for total damage done by more than one hit is:

1 hit: 5 -> 12
2 hits: 10 -> 24
3 hits: 15 -> 36
4 hits: 20 -> 48
5 hits: 25 -> 60
6 hits: 30 -> 72

The generic kobold or goblin is dead at minimum after 2 or 3 hits, and definitely dead after more than 5 or 6 hits.

To make things more precise, one can examine the probability of the total damage killing a monster after a number of hits. Let's look at the goblin with 29 hit points, using an online dice calculator. The results are:

3 hits: prob(total damage >= 29) = 23.44%
4 hits: prob(total damage >= 29) = 88.01%
5 hits: prob(total damage >= 29) = 99.83%


At a slightly higher level, the hit points of some generic monsters are:

Orc: 46 (level 3)
Hobgoblin: 39, 47 (level 3)
Skeleton: 45 (level 3)
Lycanthrope: 48 (level 3)
Doppelganger: 45 (level 3)

For our player now at level 3 with the same at-will power with a weapon doing d8 damage, the primary combat stat mod is +4 and maybe the weapon is now magic with a +1 bonus to damage. The damage per hit is d8+5. The lower and upper limits for total damage for more than one hit are:

1 hit: 6 -> 13
2 hits: 12 -> 26
3 hits: 18 -> 39
4 hits: 24 -> 52
5 hits: 30 -> 65
6 hits: 36 -> 78
7 hits: 42 -> 91
8 hits: 48 -> 104

It requires a minimum of 3 or 4 hits to kill a generic level 3 or 4 monster, while they're definitely dead after 8 or more hits.

Doing the same exercise using the online dice calculator for a generic orc with 46 hit points, we get:

4 hits: prob(total damage >= 46) = 5.13%
5 hits: prob(total damage >= 46) = 64.77%
6 hits: prob(total damage >= 46) = 98.11%


At a slightly more higher level, the hit points of some generic monsters are:

Beetle: 88 (level 8)
Behemoth: 82 (level 7)
Boar: 85 (level 6)
Troglodyte: 69, 74, 93 (level 6-8)
Zombie: 71, 88 (level 6-8)
Spider: 80 (level 7)
Shadar-Kai: 77, 86 (level 7-8)
Satyr: 80, 86 (level 7-8)
Ogre: 91 (level 8)
Foulspawn: 86, 87 (level 8)
Succubus: 90 (level 9)

For our player now at level 8 with the same at-will power with a weapon doing d8 damage, the primary combat stat mod is +5 (+1 from enough stat increases) and maybe the weapon is now magic with a +2 bonus to damage. The damage per hit is d8+7. The lower and upper limits for total damage for each hit are:

1 hit: 8 -> 15
2 hits: 16 -> 30
3 hits: 24 -> 45
4 hits: 32 -> 60
5 hits: 40 -> 75
6 hits: 48 -> 90
7 hits: 56 -> 105
8 hits: 64 -> 120
9 hits: 72 -> 135
10 hits: 80 -> 150
11 hits: 88 -> 165
12 hits: 96 -> 180

It requires a minimum of 6 or 7 hits to kill a generic level 7 or 8 monster, while they're definitely dead after 12 or more hits.

Doing the same exercise using the online dice calculator for a generic stayr with 80 hit points, we get:

7 hits: prob(total damage >= 80) = 56.41%
8 hits: prob(total damage >= 80) = 97.38%
9 hits: prob(total damage >= 80) = 98.64%

As one can see from these calculations, the number of hits using only at-will powers with d8 damage (with assumptions from magic bonuses and stat increases), goes from 5 hits to kill level 1-2 monsters, to 6 hits to kill level 3-4 monsters, to 8 hits to kill level 7-8 monsters. (The number of hits are from the first greater than 95% probability of total damage killing a monster, from the online dice calculator).

With the players having more daily powers at higher levels, in principle it should reduce the number of hits it takes to kill such higher level monsters. These calculations will be the subject of a future post.

How many hit points should a monster have?

In an older WotC article by David Noonan, it is suggestive that monsters last around 5 rounds. How exactly this is implemented, the article does not state precisely.

Perhaps this "5 rounds" means the number of times it takes to hit a monster, before it is killed by the players?

To investigate this question, we need to recall a few things.

Recall that the average number of attacks it takes to hit a monster N times is N/p, where p is the probability of hitting a monster.

Also recall that the average amount of damage done by one attack is p avg(D), where D is the damage dice (ie. such as 1d6+2) and p is the probability of hitting a monster. (avg(D) is the expectation value of D).

If a monster dies after being hit N times, it will take on average N/p attacks to kill it. So on average the number of hit points it should have, is the average number of attacks multiplied by the average amount of damage done by each attack. Assuming each attack is done by the same player with the same weapon on the same monster, the number of hit points the monster should have is:

(N/p)(p avg(D)) = N avg(D)

Interesting that the number of monster hit points in this scenario, is independent of the probability to hit the monster. N avg(D) is only dependent on the number of hits a monster will take before dying, and the average damage done by the player.