Let us assume that our victim has a probability $p$ of passing the DC 13 Constitution saving throw, and we will assume that $p$ is greater than 0 and less than 1 (that is, passing and failing the save are both possible). Let us also, for simplicity, measure damage in number of d6’s (we can convert to actual damage at the end; 1d6 has a mean damage of 3.5).

Also note that the mean (average) of a discrete probability distribution, where getting $x$ has probability $P(x)$, is given by

$$ sum_x x P(x), $$

where we sum over all values of $x$.

Before tackling the full problem, let us consider a subset of the problem. Suppose the victim has saved their way down to taking only 1d6 damage at the start of their turns. What is the average damage they will take then? We need to enumerate over each possible scenario. The victim could take 1d6 damage at the start of their turn then pass their save (with probability $p$) at the end of their turn, taking no further damage. The victim could fail one save then pass the second (with probability $(1-p)p$), taking 2d6 damage. And so on. The mean number of d6’s taken in the ‘1d6-per-round’ phase is

$$ sum_{n=0}^infty (n+1) p (1-p)^n. $$

This is an arithmetico-geometric sequence. We can solve it with some basic algebraic manipulation.

$$ sum_{n=0}^infty (n+1) p (1-p)^n = frac{p}{1-p} sum_{n=0}^infty (n+1) (1-p)^{n+1} = frac{p}{1-p} sum_{m=1}^infty m (1-p)^m = frac{p}{(1-p)} frac{(1-p)}{p^2} = frac{1}{p}.$$

In the 1d6-per-round phase, the victim takes an average of $frac{1}{p}$ lots of 1d6 damage (remember that $p$ is less than 1, so $1/p$ is greater than 1). It is simple to extrapolate that the 2d6-per-round phase will have a mean of $displaystylefrac{2}{p}$, and the 3d6-per-round phase will have a mean of $displaystylefrac{3}{p}$. Because the phases are independent, we are able to simply add together their means for the overall mean.

However, there is one tricky point with regards to the round in which the potion is consumed. If the victim drinks the potion on their turn, then the item text implies they get to make a save to reduce the damage before the start of their next turn. This means there is only a $1-p$ chance of entering the 3d6-per-round phase, so the mean damage taken from that phase is $displaystylefrac{3(1-p)}{p}$.

Let us bring everything together now. If the victim drinks the potion on their turn and fails their initial save, then they take an initial 3d6 damage plus $frac{3(1-p)}{p}$ d6 plus $frac{2}{p}$ d6 plus $frac{1}{p}$ d6 damage, for a total of

$$

3+frac{3(1-p)}{p}+frac{2}{p}+frac{1}{p} = 3 + frac{3(2-p)}{p} = frac{6}{p} mbox{d6 damage,}$$

or $displaystylefrac{21}{p}$ damage.

If we make no assumptions about the initial save, then we get the above damage with probability $1-p$ and just 3d6 damage with probability $p$, for a net result of $displaystylefrac{6(1-p)}{p} + 3p$ d6 or $10.5left(displaystylefrac{2(1-p)}{p} + pright)$ damage.

If the victim ingests the poison outside their turn (such as with a Ready action, or by having another character administer it), then the mean damage from the 3d6-per-round phase is $displaystylefrac{3}{p}$ d6. If they fail their initial save, they take $3 + displaystylefrac{6}{p}$ d6 or $10.5left(1+displaystylefrac{2}{p}right)$ damage. If we make no assumptions about the initial save, they take $frac{6}{p}-3$ d6 or $10.5left(displaystylefrac{2}{p}-1right)$ damage.

Let us consider some concrete values of $p$. The average damage dealt by the *potion of poison* is (rounded to one decimal place)…

CON Save |
$p$ |
On turn, fail first save |
On turn |
Off turn, fail first save |
Off turn |

-7 |
0.05 |
420 |
399.5 |
430.5 |
409.5 |

-6 |
0.10 |
210 |
190.1 |
220.5 |
119.5 |

-5 |
0.15 |
140 |
120.6 |
150.5 |
129.5 |

-4 |
0.20 |
105 |
86.1 |
115.5 |
94.5 |

-3 |
0.25 |
84 |
65.6 |
94.5 |
73.5 |

-2 |
0.30 |
70 |
52.2 |
80.5 |
59.5 |

-1 |
0.35 |
60 |
42.7 |
70.5 |
49.5 |

+0 |
0.40 |
52.5 |
35.7 |
63 |
42 |

+1 |
0.45 |
46.7 |
30.4 |
57.2 |
36.2 |

+2 |
0.50 |
42 |
26.3 |
52.5 |
31.5 |

+3 |
0.55 |
38.2 |
23.0 |
48.7 |
27.7 |

+4 |
0.60 |
35 |
20.3 |
45.5 |
24.5 |

+5 |
0.65 |
32.3 |
18.1 |
42.8 |
21.8 |

+6 |
0.70 |
30 |
16.4 |
40.5 |
19.5 |

+7 |
0.75 |
28 |
14.9 |
38.5 |
17.5 |

+8 |
0.80 |
26.3 |
13.7 |
36.8 |
15.8 |

+9 |
0.85 |
24.7 |
12.6 |
35.2 |
14.2 |

+10 |
0.90 |
23.3 |
11.8 |
33.8 |
12.8 |

+11 |
0.95 |
22.1 |
11.8 |
33.8 |
12.8 |

>11 |
1.00 |
— |
10.5 |
— |
10.5 |