Examining Mirror Image

[Frogreaver]

Because you can trivially prove that you’re wrong, with actual hard examples.

Take a PC of Dex 20, with Mage Armor. AC 18, Image AC 15.
The enemy makes four attacks-they roll (with bonuses) 15, 16, or 17 for all hits.
You lose all images, on average, with only a slim chance of retaining one. But they saved you no damage.

Take another PC-Dex 8, with Defensive Fighting, +3 full plate and +3 shield. AC 27, Image AC 9.
The enemy has +7 to-hit, so hits you on a 20, but your images on a 2.
The odds of your spell doing ANYTHING are pretty slim, since most attacks kill an Image without touching you.

There's one glaring problem. You've stipulated 1 specific combination of attack rolls out of like 80,000+. Your scenario reveals next to nothing probabilistically speaking.

[Valmark]

There's one glaring problem. You've stipulated 1 specific combination of attack rolls out of like 80,000+. Your scenario reveals next to nothing probabilistically speaking.

There are 80000+ scenarios like that one. And even if there weren't, that is a possible scenario that your math doesn't account for, thus proving it flawed. Besides the fact that some of that equation looks wrong as I pointed out in the previous post (I might be mistaken of course, which is why I asked).

[Frogreaver]

Can you add some explanations to those calculations? For example how did you go from "M'C + M'(1-C) + (1-M')C' + M'(1-C)" to "1"?

Good questions.

For this specific question figure out all the probabilities for everything to happen in the 1 attack case. You'll note each of those values represent the probability corresponding to 1 of those cases. Since the sum of all cases in probability always = 1 then that must =1.

You can get to "M'C+(1-M')+2M'(1-C)", but I don't see how you get 1.

Another great question.

The premise is that I can add some variable as long as I also subtract the same thing. So I simply add the variable I need to make 1 and subtract them out on the end. However, I was reviewing there and I think there is an error in that step as the 4th variable also isn't what I need that in order for it to become 1.

Good eye. I'll recalc that.

EDIT: Ended up being variable in first series at end was written incorrect. That variable was changed via copying pasting the first series down there but since the first was wrong it messed up that part as well. I've updated this. No actual changes in results.

There are 80000+ scenarios like that one. And even if there weren't, that is a possible scenario that your math doesn't account for, thus proving it flawed. Besides the fact that some of that equation looks wrong as I pointed out in the previous post (I might be mistaken of course, which is why I asked).

Expected value is pretty basic probability stuff. Comparing 2 algebraic ratios is common practice to determine what variables actually matter between them.

I'm at a loss that either of those basic concepts are being questioned. I'm at a loss that there's even a question about 1 out of 80,000 cases as if mentioning that case has any bearing on any results, especially when comparing via ratio has already cared for all the cases. I don't know what else to say on that. I'm at a complete loss that these incorrect counterpoints keep getting brought up again and again. I don't know how not to sound frustrated about that. Maybe you'll notice how I respond to actually valid points as yours about the algebra involved.

[Hellpyre]

https://docs.google.com/spreadsheets...it?usp=sharing

Here, I've put together a rough spreadsheet (sorry in advance about the output being at the bottom). You can play with both target ACs and get results 500 simulated rolls at a time. You'll also note that as PC target AC increases but MI target stays the same, the expected and actual damage will change at different rates, trending towards 25% damage taken at PC's AC = MI's AC, and increasing as the values diverge.

[HolyDraconus]

I always thought mirror image was an ok low level defensive spell, that worked until you got up to higher levels where the damage dealers aren't stupid and can just close their eyes.

[Valmark]

Can you add some explanations to those calculations? For example how did you go from "M'C + M'(1-C) + (1-M')C' + M'(1-C)" to "1"?

You can get to "M'C+(1-M')+2M'(1-C)", but I don't see how you get 1.

Funny enough, I got it wrong here. It's not "M'C+(1-M')+2M'(1-C)" but "M'C+(1-M')C'+2M'(1-C)". Forgot a C'. Still doesn't make 1 (please when you modified it say so that I don't miss it like earlier).

This also shows me one thing- why are you considering "(1-M')C' "? That is the case where the mirror image isn't targeted AND its AC is hit. Why do we care of MI's AC when it's not getting targeted?

And thinking about it, why do we consider M'(1-C) twice? (I'll double check to have copied correctly after this).

Expected value is pretty basic probability stuff. Comparing 2 algebraic ratios is common practice to determine what variables actually matter between them.

I'm at a loss that either of those basic concepts are being questioned. I'm at a loss that there's even a question about 1 out of 80,000 cases as if mentioning that case has any bearing on any results, especially when comparing via ratio has already cared for all the cases. I don't know what else to say on that. I'm at a complete loss that these incorrect counterpoints keep getting brought up again and again. I don't know how not to sound frustrated about that.

...so, according to the OP you say that MI's effectiveness stays the same regardless of AC. Your math is meant to prove that.

I have 15 AC and 13 as MIAC (which has been cast before the fight), and take, say, 7 attacks.

Let's say attacks are 14 14 14 15 15 15 15. Let's say MI triggers on the first three.

MI blocked no damage that I took, proving itself to be a waste of slots.

If my AC was, say, 14 then MI would have blocked three attacks out of the seven that hit me. It would have blocked nearly half of the damage assuming equal damage.

MI's effectiveness changed a LOT (from 0% to 100%) by varying AC by 1.

In terms of effective hp, I lost effective hp from something that normally would have done nothing.

Where is this accounted in your math?

[Frogreaver]

Funny enough, I got it wrong here. It's not "M'C+(1-M')+2M'(1-C)" but "M'C+(1-M')C'+2M'(1-C)". Forgot a C'. Still doesn't make 1 (please when you modified it say so that I don't miss it like earlier).

This also shows me one thing- why are you considering "(1-M')C' "? That is the case where the mirror image isn't targeted AND its AC is hit. Why do we care of MI's AC when it's not getting targeted?

And thinking about it, why do we consider M'(1-C) twice? (I'll double check to have copied correctly after this).

Made an update to a couple of keying errors I made in calcs. Maybe that will help you walk through it.

...so, according to the OP you say that MI's effectiveness stays the same regardless of AC. Your math is meant to prove that.

I have 15 AC and 13 as MIAC (which has been cast before the fight), and take, say, 7 attacks.

Let's say attacks are 14 14 14 15 15 15 15. Let's say MI triggers on the first three.

MI blocked no damage that I took, proving itself to be a waste of slots.

If my AC was, say, 14 then MI would have blocked three attacks out of the seven that hit me. It would have blocked nearly half of the damage assuming equal damage.

MI's effectiveness changed a LOT (from 0% to 100%) by varying AC by 1.

A few comments:

A given specific case as you describe here is nearly never going to match up with the expected value which is essentially the weighted average of all cases.

Mirror Image AC will affect the outputs just not player AC.

In terms of effective hp, I lost effective hp from something that normally would have done nothing.

Where is this accounted in your math?

The comparison ratio cares for that. (Step 3 in the formula)

[Valmark]

Adding these together and rearranging a bit we get
= (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
= (M'*C*D)*(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']
= 2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)

Now let's calculate the effective Damage you take without mirror image for the 2 attack scenario.
2. 2*C*D

Now let's calculate the effective Damage factor
3. [2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)] / [2CD]
=(M')(1+(0.5)(1-M')(C')(M''/M'-1)

As can be seen from this the effective Damage factor doesn't depend on C. Therefore, we have found that the effective Damage factor for mirror image is independent of C.

Edited: had incorrect variable accidently type in 2 places. Nothing changes with calc, was a keying error.

Whoa, wait, no. You can't add a variable in a parenthesis then subtract it from another parentesis. You need to subtract it from the same.

[Frogreaver]

Whoa, wait, no. You can't add a variable in a parenthesis then subtract it from another parentesis. You need to subtract it from the same.

Okay baby steps!

M'CD(A + B)
= M'CD(A) + M'CD(B) +M'CD(B') - M'CD(B')
= M'CD(A+B') + M'CD(B-B')

That's all I did. I just didn't baby step through it.

[JNAProductions]

To be proved valid in this instance, it has to be true ALWAYS.

Doesn’t matter how likely the opponent is to roll 15-17 only. It’s possible.

And what about the Image AC 9 and PC AC 27 situation?
That means that any given attack at +7 has a 57/80 chance of hitting an Image, but only a 1/80 chance of hitting you.

[Frogreaver]

To be proved valid in this instance, it has to be true ALWAYS.

Doesn’t matter how likely the opponent is to roll 15-17 only. It’s possible.

And what about the Image AC 9 and PC AC 27 situation?
That means that any given attack at +7 has a 57/80 chance of hitting an Image, but only a 1/80 chance of hitting you.

If someone else wants to explain why what you are doing is incorrect that's fair game. I'm tired of fending off egregious fallacies. I think from now on I'll just note when it happens and move on.

[Valmark]

Made an update to a couple of keying errors I made in calcs. Maybe that will help you walk through it.

A few comments:

A given specific case as you describe here is nearly never going to match up with the expected value which is essentially the weighted average of all cases.

Mirror Image AC will affect the outputs just not player AC.

To be proved valid in this instance, it has to be true ALWAYS.

This.

On the other point, alright but assuming I understood correctly and you put +(1-M')(1-C') where is -(1-M')(1-C')?

You really need to go through the baby steps if you want to prove your point. At least the starting point of the equation needs to show all the variables- for example, where did the "2*D*[M'*C]^2" go? Can we have the full starting equation?

[JNAProductions]

If someone else wants to explain why what you are doing is incorrect that's fair game. I'm tired of fending off egregious fallacies. I think from now on I'll just note when it happens and move on.

Your stance is, correct me if I’m wrong, that the relative AC between your PC and your images is irrelevant in terms of damage blocked.

Which means that, over, say, five attacks against our foolish AC 27 Fighter, there is normally an about...
77% chance of being hit by one of those five attacks.
Any given attack has about a 71% chance of killing an Image when there are three left.
Meaning that there’s going to be about a 20% chance that the first three attacks hit images every time, and actually hit those images, rendering the spell useless.

And again-your stance is, as far as I can tell, a UNIVERSAL statement. Meaning a SINGLE example of it being wrong disproves it.

[Frogreaver]

This.

On the other point, alright but assuming I understood correctly and you put +(1-M')(1-C') where is -(1-M')(1-C')?

There is no -(1-M')(1-C'). It's from the 4th and 7th cases in the cases I list out above the main algebra.

You really need to go through the baby steps if you want to prove your point. At least the starting point of the equation needs to show all the variables- for example, where did the "2*D*[M'*C]^2" go? Can we have the full starting equation?

M'CD got pulled out to the front leaving 2M'C. That got broken out to M'C + M'C. You'll see the frist M'C in the first long set of parenthesis. The 2nd you will see in the 2nd long set of parenthesis.

The 2 is on front of that one because that case only happens once but it's the cases when both attacks hit the PC. Thus it gets weighted for 2 attacks instead of 1 attack like the other cases.

Apologies if not enough is present for you to step through on your own. I had thought I had included enough but if you've ever posted math I'm sure you know how that goes. I'm happy to answer any questions or help with any steps that you find confusing.

[Hellpyre]

If someone else wants to explain why what you are doing is incorrect that's fair game. I'm tired of fending off egregious fallacies. I think from now on I'll just note when it happens and move on.

For what it's worth, I agree with you here. It's ludicrous to try and pull single data points to disprove statistical trends.

That said, I've found something odd in messing with the data on my own some more. Given a uniform distribution, the data agrees with you that PC AC is irrelevant to the first attack. That, in fact, MI AC is basically irrelevant as well. But as soon as I go back to a randomised distribution, the trend towards divergence reappears. It may be a data artifact. I concede to you that AC is irrelevant as far as single attacks, and apologize that my data was flawed. Running more in-depth on my copy of Excel, I still see statistical divergence on later attacks relative to the AC gap.

[Frogreaver]

Your stance is, correct me if I’m wrong, that the relative AC between your PC and your images is irrelevant in terms of damage blocked.

Then this should settle the confusion, that is not my stance.

Further, even if someone were to make such a statement, it would obviously be about the AVERAGE damage blocked per trial after running infinite trials and thus the specific actual damage blocked in any given trial would be irrelevant to disproving the claim.

For what it's worth, I agree with you here. It's ludicrous to try and pull single data points to disprove statistical trends.

That said, I've found something odd in messing with the data on my own some more. Given a uniform distribution, the data agrees with you that PC AC is irrelevant to the first attack. That, in fact, MI AC is basically irrelevant as well. But as soon as I go back to a randomised distribution, the trend towards divergence reappears. It may be a data artifact. I concede to you that AC is irrelevant as far as single attacks, and apologize that my data was flawed. Running more in-depth on my copy of Excel, I still see statistical divergence on later attacks relative to the AC gap.

Thanks, and no problem. I technically went up through case 2 attack and extrapalated that the others should also be that way. I think it's a fair extrapolation but if you are getting something quite different in a montecarlo sim then can you drop it down to a 2 attack case and see what happens? I'm curious if it will show independent of the AC gap in that case (please make sure that you are leaving the mirror image ac static and only changing the relative player ac). If it doesn't lets delve into your methodology a bit.

[Valmark]

There is no -(1-M')(1-C'). It's from the 4th and 7th cases in the cases I list out above the main algebra.

M'CD got pulled out to the front leaving 2M'C. That got broken out to M'C + M'C. You'll see the frist M'C in the first long set of parenthesis. The 2nd you will see in the 2nd long set of parenthesis.

The 2 is on front of that one because that case only happens once but it's the cases when both attacks hit the PC. Thus it gets weighted for 2 attacks instead of 1 attack like the other cases.

Apologies if not enough is present for you to step through on your own. I had thought I had included enough but if you've ever posted math I'm sure you know how that goes. I'm happy to answer any questions or help with any steps that you find confusing.

If there is no -(1-M')(1-C') why did you talk about adding a variable as long as you subtract that same variable when I pointed it out then corrected putting that? Or rather, which variable did you add/subtract?

Yeah, the 1st case I got it. Looking at it, I'm still missing where you get (1-M')(M''/M')C'. Mainly the division.

[Frogreaver]

If there is no -(1-M')(1-C') why did you talk about adding a variable as long as you subtract that same variable when I pointed it out then corrected putting that? Or rather, which variable did you add/subtract?

The full added and subtracted variable was (M'CD)(1-M')C' or if your looking in the parenthesis it will read (1-M')C' in the 2nd set of parenthesis (3rd value) and at the very end.

Yeah, the 1st case I got it. Looking at it, I'm still missing where you get (1-M')(M''/M')C'. Mainly the division.

I multiply that by M'/M' and carry the numerator M' to the front of the parenthesis. This allows the variable to keep the M'CD prefix since basically all my terms have that. (both preceded by M'CD)

[Hellpyre]

Okay, having gone down the rabbit hole and created a fixed statistical set instead of a random set, I think I see where Frogreaver is coming from here. In terms of absolute damage blocked by MI, you can model the attacks blocked by each MI in terms of how long it lasts (I.E. its personal AC) and the percentage of attacks diverted to it (I.E. the number of images remaining). These two correlate without reference to the AC of the PC caster in question. It can block attacks without bias based on the player's AC. The issue I had communicating here appears to be based entirely in different methods of eHP calculation leading to different conclusions about the final multiplier.

Please correct me if I'm wrong here, Frogreaver, but your contention here is that the number of attacks, and therefore damage, absorbed by MI disregards the AC of the PC, correct? And so the absolute eHP it offers is the same regardless of non-Dex boosts to AC.

I believe most of the rest of us are looking at what it does multiplicatively to the eHP of the PC, and seeing that the multiplier goes down as non-Dex AC grows. So we would say that it gives less effective boost relative to the non-MI eHP as player AC grows.

[Valmark]

The full added and subtracted variable was (M'CD)(1-M')C' or if your looking in the parenthesis it will read (1-M')C' in the 2nd set of parenthesis (3rd value) and at the very end.

I multiply that by M'/M' and carry the numerator M' to the front of the parenthesis. This allows the variable to keep the M'CD prefix since basically all my terms have that. (both preceded by M'CD)

Alright, the division makes sense. Now... You added (M'CD)(1-M')C' and subtracted it. You brought M'CD out and in front, yes.

This still doesn't explain why it's two +(1-M')C' instead of a +etc. And a -etc., Correct?

[Frogreaver]

Okay, having gone down the rabbit hole and created a fixed statistical set instead of a random set, I think I see where Frogreaver is coming from here.

You are getting closer but I think still some flaws in our communication.

In terms of absolute damage blocked by MI, you can model the attacks blocked by each MI in terms of how long it lasts (I.E. its personal AC) and the percentage of attacks diverted to it (I.E. the number of images remaining).

I don't think I agree with your "absolute part".

I think it's a given that the amount of time MI lasts is independent of your non dex AC. I also think that the amount of attacks diverted to it is as well. But as has been noted numerous times, not every attack directed to MI is an attack that would have hit you. So I agree that not every attack diverted to it is reduced damage. In fact I would go a step further, the higher the non-dex ac and AC disparity is the less absolute damage it is going to reduce.

However, relative damage (ie final multipler) is a different story, because while you prevent less damage you are already taking less damage. My contention (and I believe the algebra) show that those less effects are independent of non-dex ac.

These two correlate without reference to the AC of the PC caster in question. It can block attacks without bias based on the player's AC. The issue I had communicating here appears to be based entirely in different methods of eHP calculation leading to different conclusions about the final multiplier.

I don't believe there are 2 ways of accurately calculating eHP?

Please correct me if I'm wrong here, Frogreaver, but your contention here is that the number of attacks, and therefore damage, absorbed by MI disregards the AC of the PC, correct? And so the absolute eHP it offers is the same regardless of non-Dex boosts to AC.

If I'm reading you correctly, I would be saying the opposite. The absolute damage reduction lowers as non-dex ac increases but the relative (ie multiplier) stays constant regardless of non-dex ac.

I believe most of the rest of us are looking at what it does multiplicatively to the eHP of the PC, and seeing that the multiplier goes down as non-Dex AC grows. So we would say that it gives less effective boost relative to the non-MI eHP as player AC grows.

Then the problem isn't the terminology but different conclusions.

In my equation I take the eDmg with mirror image and divide by the eDmg without. If we took the inverse that would be the factor this ability can raise our eHP by provided the encounter we are in takes us right down to 0 hp (possibly a large assumption - but i'm not sure there's another way to calculate it).

Alright, the division makes sense. Now... You added (M'CD)(1-M')C' and subtracted it. You brought M'CD out and in front, yes.

This still doesn't explain why it's two +(1-M')C' instead of a +etc. And a -etc., Correct?

Some Math:

Against 2 attacks let's calculate the effective Damage you take with mirror image (this is more complicated)
There are 7 parts:
1: 2*D*[M'*C]^2
2: (M'*C*D)*(M'*[1-C])
3: (M'*C*D)*([1-M']*C')
4: (M'*C*D)*([1-M']*[1-C'])
5: (M'*[1-C])*(M'*C*D)
6: ([1-M']*C')*(M''*C*D)
7: ([1-M']*[1-C'])*(M'*C*D)

Adding these together and rearranging a bit we get
= (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
= (M'*C*D)*(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']
= 2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)

Orange is the 1 I had originally due to case 2. Red is for the added and substracted. There shouldn't be any others.

[Hellpyre]

I think it's a given that the amount of time MI lasts is independent of your non dex AC. I also think that the amount of attacks diverted to it is as well. But as has been noted numerous times, not every attack directed to MI is an attack that would have hit you. So I agree that not every attack diverted to it is reduced damage. In fact I would go a step further, the higher the non-dex ac and AC disparity is the less absolute damage it is going to reduce.

However, relative damage (ie final multipler) is a different story, because while you prevent less damage you are already taking less damage. My contention (and I believe the algebra) show that those less effects are independent of non-dex ac.

Alright. I see where that conclusion would come from. The issue we seem to be having is that as far as I can tell from working the numbers on my end, the reduction from MI is independent on a per-attack basis, but the relative modifiers from the AC gap are generally larger than the general damage modifier that the high AC provides, so they end up interdependent across the whole of the spell.

If I'm reading you correctly, I would be saying the opposite. The absolute damage reduction lowers as non-dex ac increases but the relative (ie multiplier) stays constant regardless of non-dex ac.

Then I have misunderstood you here.

In my equation I take the eDmg with mirror image and divide by the eDmg without. If we took the inverse that would be the factor this ability can raise our eHP by provided the encounter we are in takes us right down to 0 hp (possibly a large assumption - but I'm not sure there's another way to calculate it).

I do take the inverse factor there, but that's a matter of personal taste and won't affect the math any.

[Valmark]

Oh good I was starting to think that I was seeing things. Yeah I'll check anything else when I next wake up.

One last question before I turn in: did you try doing the math for more attacks then Mirror Image can stop?

After this long I think I agree that If you only take one attack in a minute, Mirror Image's effectivness doesn't care about AC.

Same thing for two. Basically, If you don't take more attacks then the three Mirror Image can stop in a minute, AC doesn't matter for it.

Math should change once you consider 4+ attacks. Because in that case MI can definitely be wasted on attacks that would do nothing while other attacks that could have been stopped hit the character.

Note: I'm using logic, I didn't do any calculation. But if you look at my earlier example with seven attacks, it proves you wrong unless we consider the distinction between 3- attacks and 4+ attacks since your calculations were based on 1 and 2 attacks.

It seemed before that you said that you assumed that the math for two attacks would hold true for many more. This is probably why then we disagree and provided several examples for that. And also why Hellpyre's found that discrepancy.

Also, small note: a single case can be used to disprove statistical trends if the trends are based on wrong basis. But I think we are using different basis, rather then Frog's math being wrong.

[Hellpyre]

Also, small note: a single case can be used to disprove statistical trends if the trends are based on wrong basis. But I think we are using different basis, rather then Frog's math being wrong.

Disprove axioms, sure. But trends really don't ever apply to single cases in-and-of-themselves.

[Frogreaver]

Oh good I was starting to think that I was seeing things. Yeah I'll check anything else when I next wake up.

One last question before I turn in: did you try doing the math for more attacks then Mirror Image can stop?

Unless I do 3 attacks and find some awesome pattern to simplify the problem then 4 is almost assuredly to large a problem for me to want to attempt. Talking probably over 50 cases to consider.

After this long I think I agree that If you only take one attack in a minute, Mirror Image's effectivness doesn't care about AC.

Well that's at least a start

Same thing for two. Basically, If you don't take more attacks then the three Mirror Image can stop in a minute, AC doesn't matter for it.

That's progress! I'm not sure why you think having the chance to run out of images would change that. I thought we already all agreed that images would expire independently of non-dex AC?

Math should change once you consider 4+ attacks. Because in that case MI can definitely be wasted on attacks that would do nothing while other attacks that could have been stopped hit the character.

Note: I'm using logic, I didn't do any calculation. But if you look at my earlier example with seven attacks, it proves you wrong unless we consider the distinction between 3- attacks and 4+ attacks since your calculations were based on 1 and 2 attacks.

Logic is good. I'm not seeing why having the potential that all images have been used is going to change anything?

I also challenge the assertion that any example proved me wrong. Everything I've seen up to now has been seriously flawed. Most common reasonable mistake was to prove that "absolute" damage reduction dropped as non dex AC increased. The mistake there was to not evaluate how much damage taken was dropping due to the higher AC.

It seemed before that you said that you assumed that the math for two attacks would hold true for many more. This is probably why then we disagree and provided several examples for that. And also why Hellpyre's found that discrepancy.

Maybe. I think if we dug more into Hellpyre's we could resolve that discrepancy. I'm still curious about what happens on his in the case of 2 attacks.

By the way I would say it's more than an assumption - it's more of a logical deduction, but of course without proof there's always the possibility that it's wrong.

[Satori01]

For a single attack it doesn't matter if any images die or not -> Also known as: C' doesn't matter in the case of taking a single attack

Or trying this a different way.
In the case of you taking 1 attack only: How much damage do you take if your image is targeted instead of you? Answer: 0

The glaring flaw in this formulation is that MI doesn't just function per hit, but rather the spells future effectiveness is determined by how many images are left.
The spells effectiveness is a function of the outcome of the three hits.

So when Mirror Image negates a "false positive" hit...a hit that would not have hit the AC of the PC, but still strikes the Image....the real harm is the spell's ability to intercept a Real future hit is diminished.

A Missile defense system that air bursts on a false target, preventing no actual Missile strikes, and now is more likely to NOT trigger when the Critical Nuclear Stike is going to land in your capital is a bad defense system.

You are ignoring this.

[Segev]

Mirror image negates 4 hits, maximum.

If your AC and the mirror image's AC are the same (or, somehow, yours is lower, which I think is impossible), it always negates exactly 4 hits, unless nothing would have hit you at all during the duration.

If your AC is higher than the mirror image's, then it is possible for mirror image to negate fewer than 4 hits, as an image disappears to an attack that would not have hit you. This probability increases the greater the difference between your AC and that of your images.

If you wish to calculate "effective hp" that mirror image is adding, you have to calculate how many hp of damage each image negates. You do this by rolling the damage dice of the attacks that hit the images instead of hitting you. This is the effective hp the image gave you.

If the hit would have missed you but still hit the image, the effective hp the image gave you is 0, because the hit would have been a miss and done 0 damage to you.

High AC is still preferable to low AC. You having higher AC than your images will still net you more protection overall than you having the same AC as your images. Mirror image never results in worse outcomes for you just by being up; the worst that can happen is all 4 images are destroyed on attacks that would have missed you, so the spell adds 0 effective hp. But as the difference in AC increases, the amount of hp mirror image effectively adds decreases. If you're counting "effective hp" of increased AC and effective hp of mirror image together, increased AC reduces the amount of effective hp that mirror image adds, but mirror image is still adding 0 or more effective hp.

[bid]

When looking at Mirror Image's impact on effective hp, the math seems to show that it is independent of AC, except for ac from dex mod.

If the hit would have missed you but still hit the image, the effective hp the image gave you is 0, because the hit would have been a miss and done 0 damage to you.

Those 2 statements are in contradiction. The second one is most likely true, ergo the first one is false.

Even if you apply statistics...
A mirror image that covers all the hit range has the effective hp of 100% of the damage.
A mirror image that covers twice the hit range has the effective hp of 50% of the damage, since half the time we are in Segev's case.

[Xetheral]

By the way I would say it's more than an assumption - it's more of a logical deduction, but of course without proof there's always the possibility that it's wrong.

Your specific first-attack and first-two-attacks cases don't appear to have the same number of degrees of freedom as the general n-attack case. I think that's why your conclusions aren't generally applicable and you're getting different results than the numerical models.

Specifically, there is a variable that can be described as "odds Mirror Image prevents no damage on a particular attack because there are no images left". When you only look at the first (up to) three attacks, this variable is zero. But the moment n>3 , that variable becomes non-trivial and changes the pattern of the results. (That variable is also non-trivial in the case where n=1, but the single attack being looked at isn't necessarily the first attack made after casting the spell.)

Instances where the number of (effective) degrees of freedom depends on the values of the inputs can be subtle and hard to spot, and their implications even less clear. So it's possible I'm off base here, but from a first look that appears to be the source of the disagreement between your approach and the numerical model.

[Chugger]

Head-spinning, but we really shouldn't be ignoring the Action Economy issue.

The question is not only "is Mirror Image worth casting at higher levels when my AC is very high" (or something like that), it's also very much "is Mirror Image worth casting in combat instead of doing something else?"

Prep rounds were normal in older versions of DnD (I started in the 70s) - but for some reason fights tend to "just start" in 5e with no prep rounds. And fights tend to be over in not too many rounds.

If it's a three round fight and you mirror image on one of those rounds, you've reduced your offensive output capacity by 33%, at least in a simplistic way of thinking about this. You could have controlled; you could have AoE'd; you could have debuffed. Usually all of those things are better than casting Mirror Image.

I don't think you can mathematically model this Action Economy issue, at least not easily. It's an instinct thing most likely.