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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
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.
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 1st2nd 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 1st2nd color
1/3 chance to get ambiguous side symptom 2nd3rd color
36,4% chance other, third color is correct. if so, then:
100% chance to get ambiguous side symptom 2nd3rd color
You find certain diagnosis in:
(36,4% * 1/3 =) 12,1%.
With 100% chance to find correctly.
You find ambiguous side symptom 1st2nd 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 2nd3rd 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
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.
I'd like to add that at least my chances dont add up to anything above 1.
Cheers! Really hope I didnt make too many errors, only double checked the math twice :S.
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.
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.