Renée's Fox Den

Fortunately, death saving throws are a lot simpler than the general case of "X of outcome 1 before Y of outcome 2" because (probability of outcome 1 + probability of outcome 2) equals 1; the presence of a "neither" case can throw a monkey into the wrench.

Since there's no "neither" outcome, the number of rolls you can possibly make is bounded; "three successes before three failures", for example, will never exceed five rolls, because after the fifth roll you're guaranteed to have at least three of one outcome or the other. That makes it possible to enumerate every possible sequence of outcomes:

SSS SSFS SFSS FSSS SSFFS SFSFS FSSFS SFFSS FSFSS FFSSS FFSSF FSFSF SFFSF FSSFF SFSFF SSFFF FFSF FSFF SFFF FFF

Let's assume that our S (success) result is rolling a 4 on 1d4, and our F (failure) result is rolling anything else. We can then work out the odds of each case, like so:

SSS = 1/64 = 16/1024 SSFS = 3/256 = 12/1024 SFSS = 3/256 = 12/1024 FSSS = 3/256 = 12/1024 SSFFS = 9/1024 SFSFS = 9/1024 FSSFS = 9/1024 SFFSS = 9/1024 FSFSS = 9/1024 FFSSS = 9/1024 FFSSF = 27/1024 FSFSF = 27/1024 SFFSF = 27/1024 FSSFF = 27/1024 SFSFF = 27/1024 SSFFF = 27/1024 FFSF = 27/256 = 108/1024 FSFF = 27/256 = 108/1024 SFFF = 27/256 = 108/1024 FFF = 27/64 = 432/1024

Adding up our target outcomes, we get 106/1024 odds of three successes before three failures, and 918/1024 odds of three failures before three successes. (If these don't add up to 1, you know you missed a case somewhere.)

Once we've got the symmetrical case (i.e., X successes before X failures), we can deal with asymmetrical cases simply by changing up which rows in the probability tree we've already constructed count as which outcome – we needn't start from scratch. For example, the odds of two successes before three failures moves all the 27/1024 rows in the above table into the "overall success" bucket, and we arrive at 268/1024 odds. Similarly, "three successes before two failures" would flip the 9/1024 rows over to the "overall failure" bucket.

To build a full Markov model, label your states by ordered pairs (S,F). The starting state is (0,0), and the absorbing states are (X,*) and (*,Y). Each non-absorbing state transitions to either (S+1,F) with probability P(S) or to (S,F+1) with probability 1-P(S). You can order your states and make a huge matrix and do all the math, but there's also software to do absorbing-state Markov calculations for all kinds of complicated systems.

With that said, we can hella cheat at this specific one using combinatorics. Breaking it up into cases makes it more work! "Roll X successes before Y failures" is the same as "Roll X or more successes on X+Y-1 dice", which is a simple sum of a binomial distribution.

To use your earlier example, we have p = 1/4, X=Y=3, which gives

The general formula for X successes before Y failures with probability p of success is then

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(Minor spoilers for Chapter Five's festival)

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He maketh me to lie down in green pastures; he leadeth me beside the still waters.