A million-to-one chance to succeed an attack? [closed]

Question

Flavor

I like when my attack rolls succeed often. D&D 5e is designed to have success rolls roughly two thirds of the time. That's too low, way too low. I think that nine times out of ten is better, much better.

I made my research and I know that the best way to succeed nine times out of ten is to have exactly a million-to-one chance¹. So to invoke that rule, my build must have exactly a million-to-one chance to hit an opponent.

Build request

Hitting as a level 1 against a level 20 opponent usually has a twenty-to-one chance (1/20) to hit: the natural crit. Hitting as a level 1 with disadvantage against a level 20 opponent has a four-hundred-to-one chance (1/400) to hit: I should roll 20 on two separate dice. Let's go down and reduce my chance to hit by adding other mechanics.

How should I build my character to have exactly a million-to-one chance to hit on an attack roll each time I want to hit? The build should have lots of dice to throw to reach that million-to-one chance, but it should reliably reach that million-to-one chance. This boils down to having a lot of dice to throw (possibly sequentially, not especially at once) to make an attack against a creature of your choice and the total chance to hit must be 1/1000000 (or equivalent: 2/2000000, 3/3000000, etc.).

Rules:

• There must be two consecutive tries to hit with 1/1000000 (or equivalent) chances.
• Use any rule from any officially published, non play-test material (meaning, take all you want from any officially published source, including PHB, DMG, VGM, XGE, adventure books, but no Unearthed Arcana)
• The character can be of any level, of any combination of classes
• Magic items are allowed
• If buffs are required, the character should use them by himself.
• The opponent to hit is of your choice from the MM or the VGM or any official adventures but wants to kill my character as soon as possible, but my character must stay alive for long enough to have at least two tries to hit with a million-to-one chance.
• (De-)Buffing rounds are allowed
• Damage dealt by the hit is unimportant
• No Wish.

---

_{1: if you know who to invoke.}

Answer 1

To solve this problem, all we need to do is have the odds multiply together to get 1 in 1 million. For this, we need the prime factorization of 1 million, and to find discreet probabilities that incorporate these factors. The prime factorization of 1 million is:

$$2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5$$

There are many dice combinations that use these factors, but I chose:

• 1 on 1d20 twice $$2 \times 2 \times 5 \times 2 \times 2 \times 5$$
• 2 on 1d10 once and 4 on 1d20 once -or- 4 on 1d20 twice $$5 \times 5$$
• 2 on 1d20 twice $$2 \times 5 \times 2 \times 5$$

One way to cause this dice combination to occur is as follow:

The Character

The character, let's call him Colt, to accomplish this is a level 12 character with the following classes:

• Fighter 2 --------------------------- (for Action Surge)
• Sorcerer (Shadow Magic) 10 ---- (for Hound of Ill Omen and spells)

...the following ability scores:

• Strength: 13 or more
• Dexterity: 9 or less
• Constitution: 28 or more
• Intelligence: 3
• Wisdom: 10 or 11
• Charisma: 20

... and the following equipment (all found in the Dungeon Master's Guide):

• adamantine armor
• potion of speed
• robe of the archmagi (attuned)
• a pistol (not proficient)

The Opponent

The opponent for our character, let's call her Olive, is a siren, the stats for which can be found in an adventure module:

the "Tomb of Horrors" adventure in Tales of the Yawning Portal

The Preparation

To prepare, Colt needs to find a siren isolated in a tank of water with at least one obstacle and some small rocks, and:

1. Don his adamantine armor
2. Use Hound of Ill Omen on himself
3. Drink his potion of speed
4. Cast confusion on himself
5. Put on his robe of the archmagi
6. Cast bestow curse on himself using a 5th level spell slot
   • Select the option:

While cursed, the target must make a Wisdom saving throw at the start of each of its turns. If it fails, it wastes its action that turn doing nothing.

7. Enter the tank

The Combat

First of all, there are four possible situations:

• Colt is both confused and cursed
• Colt is confused but not cursed
• Colt is not confused but cursed
• Colt is neither confused nor cursed

Whether or not he is cursed has no effect (he has a 4/400 chance to not be cursed, and a 4/400 chance that the curse won't affect him that turn), but we have two procedures depending on whether or not he is confused.

In either case, Colt is surprised that the Siren initiates combat as he thinks he has the utmost charm. As such, it doesn't matter whether Colt or Olive roll higher on Initiative.

Confused Colt

Olive's turn

• Seeing that Colt is confused, Olive uses a tactic:
• Swim up to Colt, use Stupefying Touch, swim behind 3/4 cover
  • Colt is guaranteed to succeed on the Concentration saving throw as the most damage Olive can deal is 21 (critical hits become normal hits)
• There is a 16/20 chance Colt is Stunned during his turn.

Colt's turn

• Make the bestow curse roll if necessary (only a 4/400 chance he has the opportunity to act).
• Make the confusion roll (only a 2/10 chance he has the opportunity to attack).
• Use the effect of haste to take the Use an Object action and doff his robe of the archmagi.
• Attack with his pistol (at Disadvantage since he is underwater).
• He must roll a 20 on both d20 to hit (1/20 twice).
• Use Action Surge.
• Take the Ready action to put on his robe of the archmagi when Olive acts next.
• Make the save against confusion (4/20 twice).

Non-Confused Colt

Olive's turn

• Seeing that Colt is not confused (a 16/400 chance), Olive recognizes his superior combat ability.
• Olive throws a rock (improvised weapon) at Colt and swims behind 3/4 cover.
  • Colt is guaranteed to succeed on the Concentration saving throw as the most damage Olive can deal is 8 (critical hits become normal hits)

Colt's turn

• Make the bestow curse roll if necessary (only a 4/400 chance he has the opportunity to act).
• Use the effect of haste to take the Use an Object action and doff his robe of the archmagi.
• Attack with his pistol (at Disadvantage since he is underwater).
• He must roll a 20 on both d20 to hit (1/20 twice).
• Use Action Surge.
• Take the Ready action to put on his robe of the archmagi when Olive acts next.
• Make the save against confusion (4/20 twice).

On the second iteration, everything is the same except Colt doesn't have the opportunity to put his robe of the archmagi back on.

The Calculation

In the version with confusion, the odds are calculated as follows:

• Chance that Colt is not stunned: $$\frac{4}{20}$$
• Chance to succeed on bestow curse saving throw: $$\frac{2}{20}\times\frac{2}{20}=\frac{4}{400}$$
• Chance to roll 09-10 on confusion 1d10 roll: $$\frac{2}{10}$$
• Chance to roll two 20s on the attack roll: $$\frac{1}{20}\times\frac{1}{20}=\frac{1}{400}$$

Multiplying these probabilities together we get:

$$\frac{4}{400}\times\frac{2}{10}\times\frac{1}{400}\times\frac{4}{20}=\frac{32}{32,000,000}$$

In the version without confusion, the odds are calculated as follows:

• Chance to succeed on confusion saving throw: $$\frac{4}{20}\times\frac{4}{20}=\frac{16}{400}$$
• Chance to succeed on bestow curse saving throw: $$\frac{2}{20}\times\frac{2}{20}=\frac{4}{400}$$
• Chance to roll two 20s on the attack roll: $$\frac{1}{20}\times\frac{1}{20}=\frac{1}{400}$$

Multiplying these probabilities together we get:

$$\frac{16}{400}\times\frac{4}{400}\times\frac{1}{400}=\frac{64}{64,000,000}$$