# D&D 5EHow long does a "save ends" spell last?

#### Xeviat

##### Hero
Hi everyone. Can someone check my math? I fiddled with the formula on this until I got it working, and it seems to match up to me doing the math long-hand.

For a 1 minute duration spell with a save at the end of each of the creature's turns, how long should a spell last with a given percent chance to fail a save?

 Failure% Average Rounds 0 0 5 0.05 10 0.11 15 0.18 20 0.25 25 0.33 30 0.43 35 0.54 40 0.67 45 0.82 50 1 55 1.22 60 1.49 65 1.83 70 2.27 75 2.83 80 3.57 85 4.55 90 5.86 95 7.62 100 10

The formula I ended up using was (chance to fail^rounds)*(1-chance to fail) to determine the percent chance of it lasting each of the given number of rounds, then it was rounds*chance all summed together to get these.

Did I get it right?

#### CapnZapp

##### Legend
1 - ((1 - chance to succeed) ^ rounds) should do it. Not sure where the extra code comes from.

Sent from my C6603 using EN World mobile app

#### Xeviat

##### Hero
1 - ((1 - chance to succeed) ^ rounds) should do it. Not sure where the extra code comes from.

Sent from my C6603 using EN World mobile app

Probably because I wanted the input to be "chance to fail", since it's more like an attack user's "chance to hit". "Chance to Affect"?

#### Goober4473

##### Explorer
Assuming they make the save at the end of their turn, and are thus affected for at least 1 round, shouldn't the minimum average rounds be 1? Or is this a calculation for extra rounds?

#### Xeviat

##### Hero
Assuming they make the save at the end of their turn, and are thus affected for at least 1 round, shouldn't the minimum average rounds be 1? Or is this a calculation for extra rounds?

Total rounds; they could succeed their first save.

#### Krachek

##### Hero
I don't have check your math,
But being paralyzed 3 rounds is equals to eternity in a 5 ed combat.

#### bid

##### First Post
The sum of 1 + 1/2 + 1/4 + 1/8 + ... = 2
This is a 50% chance of having to reroll next turn.

The general case:
S = ∑ p^i
S * p = ∑ p^i - p^0
S * (1 - p) = p^0
S = 1 / (1 - p)

If you have 19/20 of failing your save, it will take {1 / (1 - 19/20) = 20} rounds

tl;dr
Your average durations should be 20/19, 20/18, ... 20/4, 20/3, 20/2, 20/1.

#### Satyrn

##### First Post
So, I coded a test for you , doing a 1000 iterations (the save mod is 0)

DC 1: 0.0
DC 2: 0.055
DC 3: 0.097
DC 4: 0.169
DC 5: 0.228
DC 6: 0.304
DC 7: 0.428
DC 8: 0.531
DC 9: 0.679
DC 10: 0.814
DC 11: 0.98
DC 12: 1.251
DC 13: 1.426
DC 14: 1.889
DC 15: 2.31
DC 16: 3.054
DC 17: 4.203
DC 18: 5.459
DC 19: 9.131

#### Satyrn

##### First Post
And 10,000 iterations

DC 1: 0.0
DC 2: 0.0514
DC 3: 0.1131
DC 4: 0.1806
DC 5: 0.2567
DC 6: 0.334
DC 7: 0.4262
DC 8: 0.5403
DC 9: 0.6668
DC 10: 0.808
DC 11: 1.0247
DC 12: 1.2124
DC 13: 1.5228
DC 14: 1.8887
DC 15: 2.3327
DC 16: 2.8906
DC 17: 4.0015
DC 18: 5.7033
DC 19: 8.9407
DC 20: 19.4137

I'd say your math is close enough.

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