# Mirror Image Math

#### FrogReaver

##### Legend
My goal today is to find the formula to determine mirror image's accuracy benefits per round. I believe this spell may actually provide more overall defensive benefits than blur in many situations. I believe it's likely an overlooked spell.

Variables:
Let N = number of attacks per turn
Let X = Player Chance to be hit
Let Y = Mirror Image chance to be hit
Let T = Mirror Image chance to be targeted when all 3 images are up
Let T' = Mirror Image chance to be targeted when 2 images are up
Let T'' = Mirror Image chance to be targeted when 1 image is up

Where I am at. Case N=1 (for simplicity).

Turn 1: (1-T)*X
Turn 2:
Turn 3:
Turn 4:

#### Stoutstien

##### lunk
I agree that mirror image is better in most cases than blur even with the concentration factor but I've failed to find a formula for it or at least one that encompasses all the number of images at once. I've had to do each separately.
The best part of MI is that each image adds EHP equal to the damage dealt by that attack.

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#### FrogReaver

##### Legend
I agree that mirror image is better in most cases than blur even with the concentration factor but I've failed to find a formula for it or at least one that encompasses all the number of images at once. I've had to do each separately.
The best part of MI is that each image adds EHP equal to the damage dealt by that attack.
I think the formula should be fairly straight forward - it's just going to expand rather quickly. There's 4 events that can happen on any given turn (until you are out of images).

1. you are attacked, chance to target image fails, you are hit
2. you are attacked, chance to target image fails, you are missed
3. you are attacked, chance to target image succeeds, it is hit
4. you are attacked, chance to target image succeeds, it is missed

Only 1 state causes you to drop a mirror image
Only 1 state causes you to take damage
The remaining states represent no change

This may not be quite as bad as we are thinking.

#### FrogReaver

##### Legend
Well - not my goal but useful info.

I've calculated Blur's damage reduction for the nth attack coming at you.

Let C = Chance Concentration remains up through hit < 21 damage. X = Chance to be hit. N = Number of attacks. Then,

X - C^N * (X-X^2)

Unless you have extremely high concentration then blur actually loses value pretty quickly the more attacks come at you.

Someone else double check my work on this one!

#### FrogReaver

##### Legend
Well - not my goal but useful info.

I've calculated Blur's damage reduction for the nth attack coming at you.

Let C = Chance Concentration remains up through hit < 21 damage. X = Chance to be hit. N = Number of attacks. Then,

X - C^N * (X-X^2)

Unless you have extremely high concentration then blur actually loses value pretty quickly the more attacks come at you.

Someone else double check my work on this one!
This isn't working - Made a mistake here. Will need to work through the fix.

#### Paul Farquhar

##### Hero
One feature of Blur that isn't included in your calculations - it protects from Sneak Attack.

#### FrogReaver

##### Legend
One feature of Blur that isn't included in your calculations - it protects from Sneak Attack.
Which makes for a nice caveat bullet point but is complexly irrelevant when it comes to computing the normal amount of damage is will prevent.

#### Stoutstien

##### lunk
Well - not my goal but useful info.

I've calculated Blur's damage reduction for the nth attack coming at you.

Let C = Chance Concentration remains up through hit < 21 damage. X = Chance to be hit. N = Number of attacks. Then,

X - C^N * (X-X^2)

Unless you have extremely high concentration then blur actually loses value pretty quickly the more attacks come at you.

Someone else double check my work on this one!
I'll post my math once I clean it up. Currently it is a pile of chicken scratch.
Blur- works best if the users AC is moderately good against incoming attack bonus and the number of attacks per round are less than 4. Blur is at its best if the users AC is in the sold cap range(on hit is critical) and almost pointless if the target AC is so low the disadvantage doesn't help.
Things that have a meaningful impact on results:
Higher AC
Higher Con saves

Mirror image is better if only a few ,high bonus, attacks are targeting the user or numerous very weak attacks. The cross over points are about where the incoming attack has over 50% hitting you or if the weaker attacks have less than 50% chance of beating your mirror image AC.
The only way to buff this spell is a higher Dex modifier.

Both spells have the least effect facing moderate number attacks with a moderate chance to hit. Shield spell wins hands down here.

I'm going to attempt to graph all this.

#### Paul Farquhar

##### Hero
Which makes for a nice caveat bullet point but is complexly irrelevant when it comes to computing the normal amount of damage is will prevent.
I would argue that the ability to (almost) completely block massive damage matters more than a few hp either way of average damage.

#### FrogReaver

##### Legend
I would argue that the ability to (almost) completely block massive damage matters more than a few hp either way of average damage.
It matters when it can. It typically does it so rarely that it's a bullet point instead of something to factor into the formula. Get it?

#### Ashrym

##### Hero
Mirror image has a harder limit on number of attacks from which it will protect the caster. Unless the attackers really suck at hitting the AC on those images it also loses protection pretty fast.

That limit should be modelled somehow. Blur should win out long term provided solid concentration can be achieved. I haven't tried to formulate something like that since school so we'll see how long before I give up, lol.

#### FrogReaver

##### Legend
Mirror image has a harder limit on number of attacks from which it will protect the caster. Unless the attackers really suck at hitting the AC on those images it also loses protection pretty fast.

That limit should be modelled somehow. Blur should win out long term provided solid concentration can be achieved. I haven't tried to formulate something like that since school so we'll see how long before I give up, lol.
LOL. I'm finding hard to algebraically do both for blur and mirror image. It doesn't seem like the steps reduce down to a basic formula - though, it's not terrible using excel to build an algorithm to calculate though.

I find mirror image offers a lot more protection up front but decreases in effectiveness faster - whereas blur I find tends to offer less effectiveness up front (unless extremely low chances to be hit are looked at) but decreases in effectiveness slower.

Characters that dump dex and have high AC and get attacked a lot may prefer blur but I'm not convinced the difference even then is going to be great as both effects are going to trend toward taking full damage (baring con saves under +9)

#### Ashrym

##### Hero
LOL. I'm finding hard to algebraically do both for blur and mirror image. It doesn't seem like the steps reduce down to a basic formula - though, it's not terrible using excel to build an algorithm to calculate though.

I find mirror image offers a lot more protection up front but decreases in effectiveness faster - whereas blur I find tends to offer less effectiveness up front (unless extremely low chances to be hit are looked at) but decreases in effectiveness slower.

Characters that dump dex and have high AC and get attacked a lot may prefer blur but I'm not convinced the difference even then is going to be great as both effects are going to trend toward taking full damage (baring con saves under +9)
I never found the time to try, so I bolded what I find is typical IME with the caveat that blur decreasing in effectiveness more slowing is dependent on concentration ability.

#### Dausuul

##### Legend
The way I see it, blur is for eldritch knights. If you have a good Con save and a super-high AC and your primary combat function is hacking stuff to bits with a sword, blur is incredible.

If you are a primary spellcaster, however, you're much better off with mirror image. Even if blur might save you a few more hit points (haven't done the math, and I would definitely do it in a spreadsheet rather than trying to work out a formula by hand - too many variables), the fact that mirror image doesn't require concentration and blur does makes it a no-brainer. Concentration is one of a primary caster's most potent assets, and any self-defense spell should serve to protect it along with your hit points. If you have to concentrate on the defense spell, you've thrown away half its effectiveness right there.

#### Paul Farquhar

##### Hero
It matters when it can. It typically does it so rarely that it's a bullet point instead of something to factor into the formula. Get it?
It's not that rare. Especially when you consider NPC spellcasters versus a typical adventuring party, which almost always includes a rogue.

#### Paul Farquhar

##### Hero
The way I see it, blur is for eldritch knights. If you have a good Con save and a super-high AC and your primary combat function is hacking stuff to bits with a sword, blur is incredible.
Or some other way of whacking up your AC well beyond normal, such as dwarf wizard or valor bard.

#### Dausuul

##### Legend
Or some other way of whacking up your AC well beyond normal, such as dwarf wizard or valor bard.
Those are primary casters. You're still sacrificing your concentration for a defensive buff.

#### Worrgrendel

##### Explorer
My wife is currently playing a level 9 High Elf Bladesinger Wizard and has both spells. She seems to get more mileage out of Mirror Image but I think that mostly has to due with using concentration on other spells such as Haste or Animate Objects. She has used Blur to great effect but she also has a very high AC and is nigh un-hitable at times short of crits (see below). She is our party tank and also has the Sentinel feat to help with stickiness. It frustrates our DM to no end.

Base AC = 10 + 3 (mage armor) + 3 (Dex) + 1 (Cloak of Protection) = 17 + 4 (Int: Bladesong; up pretty much every fight) = 21 + 2 (Haste) = 23 + 5 (Shield spell w/reaction if needed) = 28

#### Stoutstien

##### lunk
My wife is currently playing a level 9 High Elf Bladesinger Wizard and has both spells. She seems to get more mileage out of Mirror Image but I think that mostly has to due with using concentration on other spells such as Haste or Animate Objects. She has used Blur to great effect but she also has a very high AC and is nigh un-hitable at times short of crits (see below). She is our party tank and also has the Sentinel feat to help with stickiness. It frustrates our DM to no end.

Base AC = 10 + 3 (mage armor) + 3 (Dex) + 1 (Cloak of Protection) = 17 + 4 (Int: Bladesong; up pretty much every fight) = 21 + 2 (Haste) = 23 + 5 (Shield spell w/reaction if needed) = 28
I guess your DM doesn't use dispel magic alot.

This PC could go from hero to zero very quickly.

#### Ashrym

##### Hero
Those are primary casters. You're still sacrificing your concentration for a defensive buff.
Blur might still be worth it depending on circumstances.

Mirror images can take 3 hits before they become ineffective. That's the limit I mentioned earlier. If concentration can be maintained regardless of how there's going to be a tipping point based on the number of expected attacks the spell caster takes where blur becomes the obvious choice. Whether it uses up the spell caster's concentration or not becomes moot if the extra damage after mirror image drops causes the caster to drop because concentration goes with that too.

The real question would be how often the number of attacks coming at the spell caster make blur worthwhile over mirror image. I think a spell caster limited to one or the other should go for mirror image but a wizard might prep both or swap based on expected events.