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In a recent game my opponen invested in one turn 9 chits in Bulgaria while I had already invested 2 chits against it a few turns ago. Im the same round he invested he gained a double hit about 16% then the next round another hit with 8% then one round later again a double hit with 20%. After this Bulgaria joined the Axis just one turn before Galipoli fell.
Accoring to the manual the is a 5% chance for each hit. I assume the counter chits has to be deducted from that.
Furthermore the manual says chance of a double it is 33%. He replaced the chit after the success each time.

So effective he uses 9 - 2 = 7 chits each round. This makes 35%.
For all 3 hits (with 2 double hits) the propability would be 35% x 33% x 35% x 35% x 33 % = 0,467%.

So less than a half percent. This seems to me a once in a livetime propability if the calculation is correct.

Do I miss something or was this just extreme luck.

Hi, Sorry about the long post, explanation and rant about bulgarias mobilization events incoming.
the last double (?) chit i would not consider lucky. And your stochastic looks at the one potential exact path this diplo happened but leaves out many branches in the probablity tree, therefore the percentage chance seems artificially low.

Bulgaria got "prepares for war" and was over 90% before the 2nd sucessfull diplo roll. Wether the 3rd roll hits or if it hits double or not doesnt matter at that point. They get mobilization from 3 (5?) Central power surrenders (Belgium?, Luxemburg?, Serbia, Albania and montenegro).
The 3 Balkan Surrenders already happened in our game many turns before bulgaria joined, bulgaria was joining very late as they usually join when the centrals surround nish or üskub to finish serbia off.

As 5 countries surrendered including serbia, belgrad, warsaw, nish and üskub triggered and bulgaria should have gotten serveral continous increases it is mobilization from belgrad being in central powers hands from early on in the game...
bulgaria joining only in summer 1915, a quarter of a year after serbia capitulated etc could be considered a lottery ticket but 6 out of 90

If i calculate the diplo chances you have to leave out the overkill, one diplo hit brought bulgaria in "prepare for war" range and prepare for war made them join the turn after that as they were 99%. That diplo hit afterwards coincided unecessarily bringing them if it was possible from 99% mobilization to 122% mobilization (theoretically) and therefore is completely irrelevant to the outcome and you should ignore this when juding the likelyhood events or outcomes. As the centrals completed every positive bulgaria mobilization event well before any negative event could have happened and they were continously getting 1-4% per turn per the belgrade event alone. Bulgaria would have joined with one diplo hit at being enough at the same date as they did. Wether they get double or not or a 2nd or third is irrelevant. Therefore focusing down "probabilities" like times 35% for a double hit is irrelevant to the outcome as the 65% of a normal hit would have done the same and we can safely ignore the 35+65=100 multiplier and any unnecessary rolls which dont reflect on the effective outcome.
All the events copied below, sorry for the long post.
Bulgaria got at minimum 34+turns*1% roughly 50% to their starting mob bringing them into the war or at maximum 90+turns*4% rougly 154% to their starting mob. Obviously in this game they joined super ultra late getting the minimum increase all the time. I find this much more surprising, bulgaria joined how many turns after serbia surrendered already?


The mobilization events for bulgaria:
; General surrenders to the Central Powers resulting in a
; random 3-5% increase in Bulgarian activation towards the Central Powers:
{
#NAME= General Surrenders To the Central Powers (Bulgaria->Central Powers)
#POPUP=
#IMAGE=
#SOUND=
#FLAG= 1
#TYPE= 2
#AI= 0
#LEVEL= 0
; Set global variable condition to always trigger (dummy value)
#GV= 1[1,100]
; Set link value to always trigger (dummy value)
#LINK= 0[0]
#COUNTRY_ID= 19
#TRIGGER= 100
;3-5% activation increase towards the Central Powers
#MOBILIZATION= [3,5] [1]
;General surrenders to the Central Powers forces
#CONDITION= 0 [0] [1]
}
{
#NAME= The Central Powers Capture Belgrade (Bulgaria)
#POPUP= <<TAG_15>>
#IMAGE=
#SOUND=
#FLAG= 1
#TYPE= 1
#AI= 0
#LEVEL= 0
;Set global variable condition to always trigger (dummy value)
#GV= 1[1,100]
;Set link value to always trigger (dummy value)
#LINK= 0[0]
#COUNTRY_ID= 19
#TRIGGER= 100
#DATE= 1914/08/01
;10-20% activation increase towards the Central Powers
#MOBILIZATION= [10,20] [1]
;Set variable conditions:
; 1st Line - Austria-Hungary politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 8 [1] [100] [0]
; 2nd Line - Bulgaria politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 19 [1] [0] [0]
;Serbia's capital Belgrade falls to the Central Powers
#CONDITION_POSITION= 187,94 [0,0] [1,1] [1] [0]
}
{
#NAME= The Central Powers Capture Belgrade - Continuous (Bulgaria)
#POPUP=
#IMAGE=
#SOUND=
#FLAG= 1
#TYPE= 2
#AI= 0
#LEVEL= 0
;Set global variable condition to always trigger (dummy value)
#GV= 1[1,100]
;Set link value to always trigger (dummy value)
#LINK= 0[0]
#COUNTRY_ID= 19
#TRIGGER= 35
#DATE= 1914/08/01
;1-4% activation increase towards the Central Powers
#MOBILIZATION= [1,4] [1]
;Set variable conditions:
; 1st Line - Austria-Hungary politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 8 [1] [100] [0]
; 2nd Line - Bulgaria politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 19 [1] [0] [0]
;Serbia's capital Belgrade falls to the Central Powers
#CONDITION_POSITION= 187,94 [0,0] [1,1] [1] [0]
}
{
#NAME= The Central Powers Capture Nish (Bulgaria)
#POPUP= <<TAG_15>>
#IMAGE=
#SOUND=
#FLAG= 1
#TYPE= 1
#AI= 0
#LEVEL= 0
;Set global variable condition to always trigger (dummy value)
#GV= 1[1,100]
;Set link value to always trigger (dummy value)
#LINK= 0[0]
#COUNTRY_ID= 19
#TRIGGER= 100
#DATE= 1914/08/01
;8-15% activation increase towards the Central Powers
#MOBILIZATION= [8,15] [1]
;Set variable conditions:
; 1st Line - Austria-Hungary politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 8 [1] [100] [0]
; 2nd Line - Bulgaria politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 19 [1] [0] [0]
; Central Powers unit near Nish
#CONDITION_POSITION= 190,98 [0,0] [1,1] [1] [0]
}

{
#NAME= The Central Powers Near Uskub (Bulgaria)
#POPUP= <<TAG_16>>
#IMAGE=
#SOUND=
#FLAG= 1
#TYPE= 1
#AI= 0
#LEVEL= 0
;Set global variable condition to always trigger (dummy value)
#GV= 1[1,100]
;Set link value to always trigger (dummy value)
#LINK= 0[0]
#COUNTRY_ID= 19
#TRIGGER= 100
#DATE= 1914/08/01
;8-15% activation increase towards the Central Powers
#MOBILIZATION= [8,15] [1]
;Set variable conditions:
; 1st Line - Austria-Hungary politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 8 [1] [100] [0]
; 2nd Line - Bulgaria politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 19 [1] [0] [0]
; Central Powers unit near Uskub
#CONDITION_POSITION= 189,101 [2,2] [1,1] [1] [0]
}

{
#NAME= The Central Powers Take Warsaw (Bulgaria)
#POPUP= <<TAG_26>>
#IMAGE=
#SOUND=
#FLAG= 1
#TYPE= 1
#AI= 0
#LEVEL= 0
;Set global variable condition to always trigger (dummy value)
#GV= 1[1,100]
;Set link value to always trigger (dummy value)
#LINK= 0[0]
#COUNTRY_ID= 19
#TRIGGER= 100
#DATE= 1914/08/01
;8-15% activation increase towards the Central Powers
#MOBILIZATION= [8,15] [1]
;Set variable conditions:
; 1st Line - Austria-Hungary politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 8 [1] [100] [0]
; 2nd Line - Bulgaria politically aligned with the Central Powers and not surrendered
#VARIABLE_CONDITION= 19 [1] [0] [0]
; Central Powers units east of Warsaw
#CONDITION_POSITION= 191,76 [3,3] [3,3] [1] [0]
}

I am not focussing on the outcome if Bulgaria would have joined later anyway but if it expectable in a game to get 3 diplo hits in a row where 2 of them are major hits (12-24%). The calculation therefore I have made above is around 0.5% so once
every 200 times. You did not comment if this calculation is correct.
The point is was this a lucky roll or is there a diffent calculation for diplomacy hits then stated in the manual.

Wether diplos work sometimes different than stated as research does in this game i dont know. that is a question for the developer.

To say it briefly, your math (0.5%) is still a bit off as probability is a bit more complex.

You have this scenario 1.(35*33)*(35)*(35*33) in your post
Just looking at one assumption, exactly 2 double hits and 3 diplo hits and ignoring other assumptions...
It should be this
1. (35*33)*(35*67)*(35*33)+(35*33)*(35*33)*(35*67)+(35*67)*(35*33)*(35*33) etc and this is just for your probability request of 3 hits and 2 double hits if the numbers you post are correct. If you include the whole probability of bulgaria joining than it was quite a miracle that they werent in 10 turns sooner.
The probability tree has many branches here that lead to the same outcome (2 double chits) just by your numbers. Different branches but they show in the same direction and same effective result. And if you look in a broader scope about bulgaria then the probability is even higher.