Better Diagnostic Tactics with Math!

[Lantantan]

Conclusions:
-If you want the cheapest path to a certain diagnosis, for your second color, diagnose the color unrelated to the two possible revealed layers.
-If you test the same color twice in a row and the result is twice inconclusive, there is a 86% chance the disease hides in the related second other color.
-Even if you only test a single color you can already correctly guess 63% of the times.
-Counterintuitive results at first due to relation to Monty Hall problem.
-Diagnosis+ potion inefficient for finding conclusive diagnosive.

Most of you know that a foolproof diagnosis needs 3 potions (or potion levels). But do we really need to spend 3? Can't we use the power of math to come up with better tactics? In this thread we do some math to analyse common strategies and summarize the variants. However, be aware I assume that all possible symptoms are equally likely, something I havn't had time to rigorously test yet. Please report your succes rates and sample size for any of the used tactics here!

But sometimes my diagnostic potion does nothing, in that case dont I need more than 3?
No. Sometimes it indeed SEEMS like a diagnostic potion has no effect and is thus wasted. This can happen when 2 ambiguous slots are marked and you administer another diagnostic potion of either color. If that color was not the illness color, then the potion has no more eligible slots to reveal. However, this means the illness hides for sure in the color other than the ineffective potion you tried last. So always make an effort to remember the third color you administer.


Fastest Path to Certainty:
Diagnose any color, you'll most likely get an ambiguous two color side symptom result. Now diagnose the color that is NEITHER of those ambiguous colors just revealed and you'll have the biggest chance of a certain 2 potion diagnosis.
If you get the center symptom first, diagnose either of the other colors you didnt start the diagnosis with.
This tactic has a 42,6% chance of conclusive diagnosis within 2 potions!
Improved succes rate of 26,6% per potion. Uses 2,49 potions on average.
Best for those named kids you want to cure for sure.

Two of the Same Diagnostic Color (or a single color +):
If you get an ambiguous result, cure the disease of the other color
This tactic has a 86,1% chance of correctly curing.
Improved success rate of 27,5% per potion. (But only 26,4% if you use color +)
But only has a conclusive diagnosis rate after 2 potions (or 1+) of 19,4%
Great for efficiently helping those plebs dying in the street.

Use a Single Color:
If you get a side ambiguous result, cure the disease of that other color, if you get ambiguous center result, guess either of the other 2 diseases.
This tactic has a 63,8% chance of correctly curing.
Improved succes rate of 30,5% per potion.
Only tiny 8,3% conclusive diagnosis chance.
Great return rate for when you're really desperate.

Single Color Complication:
The same as single color but additionally you add a second diagnostic potion if you get the ambiguous middle symptom. In that case, diagnose another color, if the result's ambiguous with the first color, cure the second. If it's ambiguous with the third, cure the third.
This tactic has a 75,9% chance of correctly curing.
Improved succes rate per potion of 32,5% per potion. using 1,31 potions on average.

[Lantantan]

Just two of the same diagnostic color:

Naively you might think, Oh because the chance our initial color pick was correctly is 1/3, the total chance to succeed is thus (1/2 * 1/3 + 1 * 2/3) = 5/6 = 83,3%. However, if your first color pick didn't yield a certain diagnosis, the odds increase that it's the second disease. Much like the Monty Hall problem. Because of this, we expect slightly better odds. Here's the breakdown:

initial color guess was correct chances:
(1/3 * 1/4 =) 1/12 = 8,3% chance of conclusive result outright. Hurrah!
(1/3 * 2/4 =) 2/12 chance of getting ambiguous side piece.
(1/3 * 1/4 =) 1/12 chance of getting center symptom.

initial color guess was incorrect chances:
(2/3 * 2/3 =) 4/9 chance of getting ambiguous side piece.
(2/3 * 1/3 =) 2/9 chance of getting middle symptom.

Which correctly adds up to 1. So how does the math change if we didn't get a conclusive result at the start? If we get an ambiguous side piece, we expect a very high chance for the correct disease to be the other one, much like in the monty hall problem.

Test result is an ambiguous side piece:
(2/12 + 4/9 =) 11/18 chance of occuring, so
(18/11 * 4/9 =) 72,7% chance to be the other disease.

Test result is ambiguous middle symptom
(1/12 + 2/9 =) 11/36 chance of occuring, so
(36/11 * 2/9 =) 72,7% change to be EITHER of the other diseases.


Now we diagnose same color again.

2/12 * 12/11 = 18,2% chance original color is correct and result was side. If so, then:
2/4 = 50,0% chance for unambiguous detection yay!
2/4 = 50,0% chance for second ambiguous result (center).

1/12 * 12/11 = 9,1% chance original color is correct and result was center. If so, then:
1/3 = 33,3% chance for unambiguous detection yay!
2/3 = 66,6% chance for second ambiguous result (side).

4/9 * 12/11 = 48,5% chance original color was wrong and result was side. If
100% chance for second ambiguous result (center).

2/9 * 12/11 = 24,2% chance original color was wrong and result was center.
100% chance for second ambiguous result (side).

So chance for unambiguous detection is:
0,182/2 + 0,091/3 = 12,1%
So chance for a second ambiguous detection is:
100% - 12,1% = 87,9%

And when detection is ambiguous, the correct disease is the other color in
1/0,879 * (48,5% + 24,2%) = 82,7% of those cases.

So to sum it all up:
8,3% chance of conclusive result at first potion.
91,7% chance second same potion is needed.

12,1% chance of conclusive result at second potion.
87,9% chance of second ambiguous measurement.
In that case, disease is correctly in second color 82,7% of the times.

Total cure rate of this tactic is:
8,3% + (91,7% * (12,1% + (87,9% * 82,7%) ) ) = 86,1%


But what about our diagnostic efficency, rate improvement per potion? Because we can be correct on the first try, we only use (0,083 * 1 + 0,917 * 2 =) 1,92 potions. So efficiency is (86,1% - 33,3%)/1,92 potions = 27,5% per potion

[Lantantan]

Only use a single color diagnostic

initial color guess was correct chances:
(1/3 * 1/4 =) 1/12 = 8,3% chance of conclusive result outright. Hurrah!
(1/3 * 2/4 =) 2/12 chance of getting ambiguous side piece.
(1/3 * 1/4 =) 1/12 chance of getting middle symptom.

initial color guess was incorrect chances:
(2/3 * 2/3 =) 4/9 chance of getting ambiguous side piece.
(2/3 * 1/3 =) 2/9 chance of getting middle symptom.

Which correctly adds up to 1. So how does the math change if we didn't get a conclusive result at the start? If we get an ambiguous side piece, we expect a very high chance for the correct disease to be the other one, much like in the monty hall problem.

Test result is an ambiguous side piece:
(2/12 + 4/9 =) 11/18 chance of occuring, so
(18/11 * 4/9 =) 72,7% chance to correctly be the other disease.

Test result is ambiguous middle symptom
(1/12 + 2/9 =) 11/36 chance of occuring, so
(36/11 * 2/9 =) 72,7% change to be EITHER of the other diseases.
You'd still have a 50/50 guess, so final effectiveness is 36,4%


So finally, the effectiveness of this tactic is:
1/12 you cure outright +
11/18 chance for a situation in which you guess correctly 72,7% of the times +
11/36 chance for a situation in which you guess correctly 36,4% of the times =

63,8% you cured the correct disease.

Which, as expected, is better than the 33,3%. You get 30,5% succes rate extra per potion with this method.

[Lantantan]

Single color complication:

We're losing a lot of efficiency when we get that nasty center symptom, so what if we only use a second diagonistic potion when we get that symptom?

Then we need two colors in 11/36 = 30,6% of the cases.
But which second color do we use?
Arg, Not brain can't handle all the branching.


Diagnose other color.
27,2% chance original color is correct. if so, then:
100% chance to get ambiguous side symptom 1st-2nd color

36,4% chance second color is correct. if so, then:
1/3 chance to get certain diagnosis
1/3 chance to get ambiguous side symptom 1st-2nd color
1/3 chance to get ambiguous side symptom 2nd-3rd color

36,4% chance other, third color is correct. if so, then:
100% chance to get ambiguous side symptom 2nd-3rd color

You find certain diagnosis in:
(36,4% * 1/3 =) 12,1%.
With 100% chance to find correctly.

You find ambiguous side symptom 1st-2nd color in
(36,4% * 1/3 + 27,2% =) 39,3%.
Then (1/0,393 * 27,2% =) 69,1% chance 2nd color hides the illness

You find ambiguous side symptom 2nd-3rd color in
(36,4% * 1/3 + 36,4% =) 48,6%.
Then (1/0,486 * 27,2% =) 75,0% chance 3rd color hides the illness.

So finally, the succes rate during the 30,6% of central symptom cases of this method is:
12,1% * 100% + 39,3% * 69,1% + 48,6% * 75,0% = 75,7%


But we're not there yet, now we need to combine all the procedure step rates. So finally, the effectiveness of this tactic is:
8,3% you guess outright +
11/18 chance for a situation in which you guess correctly 72,7% of the times +
11/36 chance for a situation in which you guess correctly 75,7% of the times =

75,9% you cured the correct disease.
You use 1 * 69,4% + 2 * 30,6% potions = 1,31 potions.
Improved succes rate per potion = 32,5% per potion

[Lantantan]

Fastest Path to Certainty:

After using a single potions:
Initial color guess was correct chances:
(1/3 * 1/4 =) 1/12 = 8,3% chance of conclusive result outright. Hurrah!
(1/3 * 2/4 =) 2/12 chance of getting ambiguous side piece.
(1/3 * 1/4 =) 1/12 chance of getting middle symptom.

Initial color guess was incorrect chances:
(2/3 * 2/3 =) 4/9 chance of getting ambiguous side piece.
(2/3 * 1/3 =) 2/9 chance of getting middle symptom.


Test result is an ambiguous side piece:
(2/12 + 4/9 =) 11/18 chance of occuring


If we again use the same color as the first, we'll only get a conclusive diagnosis in:
(100% - 61,1%) * 2/4 = 19,5% of the cases.
If we use the other ambiguous color, we'll get conclusive in:
61,1% * 2/4 = 30,5% of the cases.
If we use the 3rd unrelated color, we'll get conclusive in:
1/2 = 50,0% of the cases! Noice.


Test result is the ambiguous middle symptom
(1/12 + 2/9 =) 11/36 chance of occuring, so
(36/11 * 2/9 =) 72,7% change to be EITHER of the other diseases.

Picking the first color again gives a conclusive result in only:
(100% - 72,7%)*1/3 = 9,1% of the cases.
Picking either of the other colors gives a conclusive result in:
72,7% * 1/2 * 1/3 = 12,1%, bummer.

So using the best path gives:
1/12 = 8,3% chance for consclusive result after single potion.
(11/18 * 50,0%) + (11/36 * 12,1%) = 34,3% chance at the second potion.
So (8,3% + 34,3% =) 42,6% chance of conclusive diagnosis within 2 potions!

Diagnostic Efficiency is:
0,083 * 1 + 0,343 * 2 + (1 - 0,083 - 0,343) * 3 = 2,49 potions.
100% - 33,3% = 66,3% improvement so
26,6% Improvement per potion.

[Ray tracing is faster]

https://www.youtube.com/watch?v=msDuNZyYAIQ

[Lantantan]

Hahahaha yeah I wanted to avoid that by hiding my math in the other posts.
I'd like to add that at least my chances dont add up to anything above 1.

[Lantantan]

Oef, If I had done this optimization ingame I'd have spend 2 days just calculating. That mathmatician girl really should have done this for us. I think I've fixed most of my bugs and the results make intuitive sense. Maybe someone should turn this into a guide.

[anaphylactic god]

Originally posted by Some kind of mech:

https://www.youtube.com/watch?v=msDuNZyYAIQ

NUMBERS DONT LIE

[Lantantan]

I've been using my fastest path to certainty method during the last 4 game days but I was too stressed to keep notes. It did feel like the early reveal chance was somewhere between 20% and 40%. Anyone manage to take some ingame notes? I'll get back in when we get our bachelor.

[Mar]

Love this. Awesome, thank you all for sharing your knowledge

[Lantantan]

Originally posted by Mar:

Love this. Awesome, thank you all for sharing your knowledge

Cheers! Really hope I didnt make too many errors, only double checked the math twice :S.

[Waismuth]

ok so is it better to use normal or plus tinctures to show symptoms? And what about pain, does it give you better chances at revealing symptoms if pain is low?

[Lantantan]

Originally posted by Waismuth:

ok so is it better to use normal or plus tinctures to show symptoms? And what about pain, does it give you better chances at revealing symptoms if pain is low?

For absolute certainty diagnoses, 2 normal tinctures are more efficient (40% certain after 2) than a single plus. A single+ will only rarely (20%) yield a definitive diagnosis.

The answer depends a little because both diagnosis modes have different opportunity costs. Potion+ can also be drunk for stat bonuses so are generally more valuable, but if you're selling herbs or potions, 2 normal pots tend to be more valuable.

I must confess I havn't really noticed or paid attention to the effect of pain. All I noticed is that pain blocks further administration of anything if it is full.

[Lara Croft]

I believe that the tutorial indicated that lower pain level increases the chance of successful diagnosis.

[Lantantan]

Originally posted by Snorlax, the grapefruit pokémon:

I believe that the tutorial indicated that lower pain level increases the chance of successful diagnosis.

I mean, it does, because at the maximum pain you simply cant try more diagnostic potions. If it's more nuanced than that, it could only significantly impact above math when the illness color isn't locked in from the start. I really can't imagine they'd code it that way.
They might theoretically only determine the illness color at first potion administering and scew that result based on pain. In that case the 8,3% single potion diagnosis chance could be modified. My experience from a single playthrough makes me say that number feels about right, though it could well be different. Guess I'll keep some records during my next playthrough.