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In this Unearthed Arcana, they give rules so that players can make all the rolls, rather than the DM sometimes rolling for enemies. In one section, there are specifically rules for converting saving throw bonuses into an equivalent DC, which the players roll against, rather than the DM. However, after doing some math, I think they've made a mistake.

They say that you convert a Saving Throw into a Saving Throw Check by adding 11 to the defender's saving throw modifiers, and using that as the DC for the check. The player then rolls against this DC, adding their spellcasing ability modifier and proficiency bonus. However, according to the following math, this conversion does not produce the same results.

In this math, I compare the chances of a player succeeding on a saving throw check, using the rules in the UA article, and of a monster failing its saving throw using the standard PHB rules. If the math is correct, both of these results should have the same chance.

These are the formulas I used.

Chance to succeed Saving Throw Check = (20 - DC + 1 + proficiency + casting mod) / 20

Chance to FAIL a Saving Throw = (DC - 1 - save proficiency - save mod) / 20

Assuming a proficiency bonus of +2, +0 for all ability mods, and no save proficiency:

Monster's Saving throw Check DC = 11 + save mod = 11

Chance to succeed a saving throw check DC11: (20 - 11 + 1 + proficiency + casting mod) / 20 = 12/20 = 60%

Player's Saving throw DC = 8 + proficiency + spellcasting ability mod = 10

Chance for monster to fail saving throw DC10: (10 - 1 - save proficiency - save mod) / 20 = 9/20 = 45%

If this were a valid conversion, both formulas would result in the same chance of success for the player, and failure for the monster respectively, however, they are off by 15%. This suggests that the unearthed Arcana's rules are not a valid conversion.

Are the Unearthed Arcana rules wrong, or is my math wrong?

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    \$\begingroup\$ We did this math before 5E came out and the answer was that you should use 14 as your "magic number" to make the probabilities match (with the caveat that being the one that rolls the dice sometimes also comes with added benefits like advantage/disadvantage or reroll mechanics). (14-11)/20 = 0.15, which is where that 15% error that everyone is noticing comes from. \$\endgroup\$ – Alex P Nov 28 '16 at 21:52
  • \$\begingroup\$ Ooh... those might be duplicates.... \$\endgroup\$ – nitsua60 Dec 14 '16 at 17:08
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UA is wrong: a simple example shows it so.

Wizwiz, a level 5 wizard with INT 18 casts Charm Person on Barbar, a barbarian with WIS 8.

PHB Rules

Wizwiz's spell save DC is (8 + proficiency bonus + INT mod) = 15. Barbar's WIS save modifier is-1. If Barbar rolls a 16-20 he resists the charm. Barbar resists 25% of the time.

UA Rules

Wizwiz's save check modifier is (DC - 8) = +7. Barbar's resistance DC is (11 + WIS save mod) = 10. Wizwiz's spell takes full effect on a roll of 3-20; Barbar resists on a roll of 1 or 2. Barbar resists 10% of the time.

Therefore there exists a case where the results are not identical, and UA rules do not faithfully reproduce PHB rules.

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UA is wrong

It doesn't reproduce faithfully the results of spells adjudicated by PHB rules.

I get the same error-by-three as you. (So does reddit, for what it's worth.)

Conventions:

"castmod": caster's proficiency bonus+casting ability modifier. (Yes, 8+castmod is just the spell save DC, but we need to break it out for algebra)

"savemod": target's proficiency bonus if proficient+targeted ability modifier

\$\text{d}20^+\$: result of the roll of a d20 on which a 20 would be the most beneficial result to the caster

\$\text{d}20^-\$: result of the roll of a d20 on which a 1 would be the most beneficial result to the caster

under normal rules a caster has full effect when $$\text{d}20^- + \text{savemod} < 8 + \text{castmod}$$ or, flipping sides, $$8 + \text{castmod} > \text{d}20^- + \text{savemod}$$

under UA rules a caster has full effect when $$\text{d}20^+ + \text{castmod} >= 11 + \text{savemod}$$ (see the '>=' vs '>'? It has to do with who wins ties.)
Note that being \$>= n\$ is the same as being \$> n-1\$, in the world of integers. UA rule becomes $$\text{d}20^+ + \text{castmod} > 10 + \text{savemod}$$ Here's the fun part: the result of a \$\text{d}20^+\$ roll is the same as 21 minus the result of a \$\text{d}20^-\$ roll. Algebraically, I'm saying \$\text{d}20^+ = 21 - \text{d}20^-\$. Substituting, $$(21 - \text{d}20^-) + \text{castmod} > 10 + \text{savemod}$$ add that \$\text{d}20^-\$ roll to both sides

$$21 + \text{castmod} > 10 + \text{d}20^- + \text{savemod}$$ and subtract 10 from both sides $$11 + \text{castmod} > \text{d}20^- + \text{savemod}$$

So I'm saying that the caster has full effect when:

\$8 + \text{castmod} > \text{d}20^- + \text{savemod}\$ under the normal rules, but when
\$11 + \text{castmod} > \text{d}20^- + \text{savemod}\$ under the UA variant.

Which is a difference of 3-out-of-20, the same as your 15%.

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