Death Saving Throw Success and Failure

In Dungeons and Dragons, a character who loses all of their hit points is knocked unconscious. After that, they roll dice to see if they die permanently or if they stabilize. Each turn, they roll a 20-sided die (a d20). A 10 or higher is a success, and a 9 or lower is a failure. There are two special cases: rolling a 1 counts as two failed rolls, and rolling a 20 is an automatic success, stabilizing you with no further rolls needed. Three successes stabilizes you, and three failures kills you permanently. The probability of success in any individual roll is 55%, but what is the probability of success on the whole series of rolls? To answer this, I’ll do a quick simulation.

Because we’re dealing with random numbers, I will set.seed() to ensure replicability.

set.seed(20230908)

For a single trial:

failures <- 0
successes <- 0
while (failures < 3 && successes < 3) {
  roll <- sample(1:20, 1)
  print(paste("Roll:", roll))
  if (roll == 1) {
    failures <- failures + 2
  } else if (roll < 10) {
    failures <- failures + 1
  } else if (roll == 20) {
    successes <- 3
  } else {
    successes <- successes + 1
  }
  print(paste("  Successes:", successes))
  print(paste("  Failures:", failures))
}

[1] "Roll: 18"
[1] "  Successes: 1"
[1] "  Failures: 0"
[1] "Roll: 11"
[1] "  Successes: 2"
[1] "  Failures: 0"
[1] "Roll: 20"
[1] "  Successes: 3"
[1] "  Failures: 0"

Brilliant, I’m alive!

But now we want to repeat that many, many times to calculate the probability of success. I’m going to populate a vector called outcomes with whether each trial resulted in a success – TRUE means three successful saves, FALSE means three failed saves.

n <- 1e6

outcomes <- replicate(n, {
  failures <- 0
  successes <- 0
  while (failures < 3 && successes < 3) {
    roll <- sample(1:20, 1)
    if (roll == 1) {
      failures <- failures + 2
    } else if (roll < 10) {
      failures <- failures + 1
    } else if (roll == 20) {
      successes <- 3
    } else {
      successes <- successes + 1
    }
  }
  successes == 3
})

mean(outcomes)

[1] 0.595217

After one million repetitions, it looks like the probability of success is about 59.5%. Better than 50-50, but still…not great when you’re talking life and death.