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%