# How many rounds is persistent damage expected to last?

If you do not spend any actions to deal with it, persistent damage ends if you succeed on a DC 15 flat check.

What is the expected duration? (as calculated from the chance of ending it)

# 3+1/3 turns

On your first turn, you always take damage ($$\p=1\$$). The probability of failing a DC 15 flat check is $$\0.7\$$. So on any turn after the first, you have $$\0.7\$$ chance of taking damage, if you took damage the last turn. So that makes the overall expected value:

$$S = 1 + 0.7 + 0.7^2 + 0.7^3 + … \text{(ad infinitum)}$$

An easy way to calculate that is to observe that:

$$S \times 0.7 = 0.7 + 0.7^2 + 0.7^3 + 0.7^4 + … = S - 1,$$

from which (by subtracting $$\S\$$ from both sides and multiplying by $$\-1\$$) we get:

$$S \times 0.3 = S \times \frac{3}{10} = 1$$

and thus:

$$S = \frac{10}{3} = 3 + \frac13.$$

# 3.333 rounds

Based on geometric distribution, the average number is 1/p, where p the chance of success.

You have 30% chance to succeed on a DC 15 flat check (6 in 20). So you have to try 20/6 times on average to succeed, 3.333 rounds.

You do not take the next damage when you succeed, so we we need to subtract 1, but we also add 1 as you take damage even before the first check.

• Following your reasoning, if you had a 20/20 chance of success, you would have to try 20/20 times to end it, which is 1 turn. Add one for the first turn and you get 2. Something is not right here. May 23, 2021 at 8:29
• Good point, if you succeed, you avoid the damage, not just stop the sequence. May 23, 2021 at 9:17
• @Szega, fixed, thanks May 23, 2021 at 12:28
• "You do not take the damage when you succeed..." This is confusing b/c the damage happens before the check. I understand what you're trying to say, but I feel like it would be better noted that you take damage before the check, then you might protect yourself from the damage starting the next round. May 23, 2021 at 13:03
• @SpearCarrier.no2 No, that is not true, you take the damage before you roll to shake off the condition and it just says "If you succeed, the condition ends." Nothing about retroactively negating damage. May 23, 2021 at 16:09