What is the chance of failing all three saves of the spell Flesh to Stone?

Question

The spell flesh to stone is a powerful way to eliminate a target. However, it calls for multiple saving throws, a complication which makes determining the chance of success or failure challenging. To quote,

You attempt to turn one creature that you can see within range into stone. If the target's body is made of flesh, the creature must make a Constitution saving throw. On a failed save, it is restrained as its flesh begins to harden. On a successful save, the creature isn't affected.

A creature restrained by this spell must make another Constitution saving throw at the end of each of its turns. If it successfully saves against this spell three times, the spell ends. If it fails its saves three times, it is turned to stone and subjected to the petrified condition for the duration. The successes and failures don't need to be consecutive; keep track of both until the target collects three of a kind.

If I cast flesh to stone on someone and my concentration is not broken, what is the probability that the target is petrified as a function of how likely they are to fail an individual save?

We'll assume that the first failure is included in the tally of three (although you may answer the other case as well if you want).

Answer 1

Let the probability of the target passing its save be \$p\$. It will be easiest to enumerate the ways a creature can successfully avoid turning to stone.

To avoid turning to stone, a creature must either make the first save, or fail the first save then get three successes with no more than one more failure. The last roll must be a success. This gives us five possible outcomes where the target is not petrified:

S, FSSS, FFSSS, FSFSS, FSSFS.

(S=Success, F=Failure)

The probability of these scenarios are

$$ P(S) = p, $$

$$ P(FSSS) = (1-p)p^3, $$

$$ P(FFSSS) = P(FSFSS) = P(FSSFS) = (1-p)^2p^3. $$

The probability of not turning to stone is the sum of these probabilities,

$$ P({\rm Not\ petrified}) = p + (1-p)p^3 + 3(1-p)^2p^3 = p + 4p^3 - 7p^4 + 3p^5. $$

The probability of turning to stone is the complement of this,

$$ P({\rm Petrified}) = 1 - P({\rm Not\ petrified}) = 1 - p - 4p^3 + 7p^4 - 3p^5. $$

Converting \$p\$ to the target number on a straight d20 roll, we get the following plot. (The straight grey line represents the probability of failing on a single saving throw.)

When the number you need to roll (the spell save DC minus your Constitution save modifier) is 13 (or p≈0.4), you have a 50% chance of being petrified. If you have advantage on the rolls, the 50% mark happens when the target number is between 16 and 17. If you have disadvantage on the rolls, the 50% mark happens when the target number is between 8 and 9.

The multiple saves required for flesh to stone skews the odds slightly in favour of the target compared to a single-save spell, especially when the roll is easier.

Answer 2

Target	Expectation of Petrification	Expectation of Life as Usual
20	94.95%	5.05%
19	89.67%	10.33%
18	83.98%	16.02%
17	77.82%	22.18%
16	71.19%	28.81%
15	64.14%	35.86%
14	56.78%	43.22%
13	49.25%	50.75%
12	41.72%	58.28%
11	34.38%	65.62%
10	27.41%	72.59%
9	20.99%	79.01%
8	15.30%	84.70%
7	10.45%	89.55%
6	6.54%	93.46%
5	3.62%	96.38%
4	1.64%	98.36%
3	0.52%	99.48%
2	0.07%	99.93%

There's a discrete solution I used to generate the above table that takes advantage of a counting trick that is pretty computationally efficient and also pretty compactly expressed via dyce¹. You can see my attempt at a well-commented walkthrough and play around with it in your browser: [source]

I've attached some anydyce² "burst" graphs generated from that notebook to help give a "feel" for outcomes based on target number.

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¹ dyce is my Python dice probability library.

² anydyce is my visualization layer for dyce meant as a rough stand-in for AnyDice.

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