Act I with random choices - Probabilities

[Mirage_GSM]

Somebody asked about the probabilities for getting to the different Act I endings by purely random choices in the Ask thread. I took a shot at calculating the odds this afternoon, but I made a few mistakes, and since Aura suggested:
[quote=”Aura”]If you guys want to work it out (possibly actually worthwhile), take it to a new thread.[/quote]
I’ll do that.
Not sure if it’s really worthwhile, but it’s an interesting problem anyway, so let’s go step by step:

Leading up to “Lunch Evolution Theory” there are five choices. Four of them are binary and one of them has three options. This equals 48 different combinations.
You can set up to three flags per girl, and you need two of them for the option of the path in question to appear in LET.
The probability that you will unlock the choice for Shizune path only is 13 out of 48 or 27,08%.
The probability that you will unlock the choice for Lilly/Hanako path only is the same.
The probability that you will unlock both choices is 7 out of 48 or 14,58%.
The probability that you will unlock neither choice is 15 out of 48 or 31,25%.

Here’s where things become a bit tricky since the next choice point has a variable number of options:
If you unlocked one choice you will have two options with a chance of 50% to set the flag for the respective girl’s path. 27,08% *0,5 -> 13,54% for either Shizune or Hanako/Lilly.
If you unlocked both choices you will have three options with a chance of 33% each to set the flag for Shizune respectively Lilly/Hanako path. 14,58% *0,33 -> 4,86% for either Shizune or Hanako/Lilly.
If you unlocked neither choice you won’t be able to set any flag.
Add the probabilities, and you get 18,40% each for Shizune and for Lilly/Hanako path.
The remaining 63,19% mean that you set the flag for neither path and won’t be able to get their endings.

Next is “Cold War”, the confrontation between Shizune and Lilly. (“Waylay” comes in between, but since this is all multiplication and you always have to go through both choices the order doesn’t really matter here. I will get back to this choice later on.) It is a binary choice. Between Lilly and Shizune and will enable you to set the second flag. Basically it halves the chances that you will get on Shizune respectively Lilly/Hanako path to 9,2% each.
The probability that both routes will be locked for you is now 81,60%. I’ll call this the Emi/Rin Path.

From here on out I’ll examine those three cases separately.
Emi/Rin Path
Here the next (or previous if you will) relevant choice is “Waylay”. It is a binary choice. To get to Emi’s end, you have to pick “correctly” which will take you to “Exercise” where you’ll have to pick correctly again which makes the probability to reach Emi’s end this way 81,60% *0,25 -> 20,40%.
If you pick the “wrong” choice in “Waylay” you get to “Mind your Step” instead followed by “Creative Pain”. Both are binary “Kenji-choices” i.e picking the wrong one will send you on a one way path to the Deep End.
Consequently you get the following result for Emi/Rin path:
Kenji: 45,9%
Emi: 20,40%
Rin: 15,3%
which adds up to the 81,60% for this path

I made a mistake here in my earlier calculation, because I simply forgot the second Kenji-choice in “Creative Pain” so a significant portion of Rin’s total from my earlier calculation should be Kenji’s.

Lilly/Hanako Path
There is a 50% chance you set the Emi-flag in “Waylay”. If you did you also get to see the choice in Exercise to get on Emi route. The probability for this is 9,2% *0,25 -> 2,3%
If you didn’t chose Emi in “Wayay” you’ll get one chance to get on Kenji’s path in “Mind your Step” and a final choice between Hanako and Lilly. The same is true if you don’t choose Emi again in “Exercise”.
I think not considering the possibility to get back on Emi’s route from here was another mistake I made earlier. (I forgot to save the excel sheet I used for the calculations^^°)
So the results for Lilly/Hanako Path are:
Kenji: 3,45%
Emi: 2,3%
Lilly/Hanako: 1,73% each
which adds up to the 9,20% for this path

Shizune Path
*deep breath*
In this path there is also the possibility that you set the Emi-flag in “Waylay” (4,60%). If you did not you’l get directly to the choice in “No Recovery” which will send you either towards Kenji’s end or Shizune’s (2,3% each).
If you did you will see the choice in “Exercise” with a 50% chance to get to “No Recovery” again binary choice between Kenji and Shizune, this time only 1,15% each.
Otherwise you’ll get to “Slow Recovery” with a 50% chance for the Kenji End (1,15%) and 50% for another choice between Emi and Shizune (0,58% each).
Adding all this up we get:
Kenji: 4,60% (2,30%+1,15%+1,15%)
Shizune: 4,03% (2,30%+1,15%+0,58%)
Emi: 0,58%
which adds up to the 9,20% for this path

Total probabilities for Act 1 are:
Kenji: 53,95%
Emi: 23,28%
Rin: 15,30%
Shizune: 4,03%
Lilly: 1,73%
Hanako: 1,73%
which adds up to 100%

Notguest did a similar calculation before with slightly different results. The post can be found here. Since he didn't post how he arrived at those results, I can't tell if or where he made a mistake.

If somebody finds a mistake in my calculation, I will gladly correct it and give credit to whoever found it. It is a pretty complex problem after all…

[BMFJack]

Holy balls. More than a 50% chance of manly picnic. Makes me wonder how I didn't get it my first time.

I find it a little odd that the Lilly/Hanako percentages are so low, but it does make a certain kind of sense. I had to try twice for Lilly, but got everyone else on the first try. Of course, once I knew how to get Lilly that brings Hanako in as well. A friend of mine said it took him several tries to get Hanako.

I've always heard people say that Emi is the easiest to get and that most people get Emi their first time, but I don't really follow their logic. To get Emi you have to specifically disobey the Nurse and take a huge risk by pushing yourself too hard. I got Rin my first time.

[AaronIsCrunchy]

More than a 50% chance of a manly picnic, and I had to ask on here after 4/5 months how to get on it! As a statistics student, I do want to dissect this a bit, but I might leave it till tomorrow. Interesting read though!

[SpunkySix]

Gosh, no wonder I ended up with Emi even though I went in thinking about going for Hanako first. Then again, that has more to do with the context of the choices than chances, but still. Other than falling off of the roof, that's pretty far above the others in terms of numerical probability.

[Notguest]

Since we got different results, I decided to double check and redo the calculations as exhaustively as possible. Since I spent two pages plotting out the flow charts, I'm pretty confident. Note that when I say the path doesn't branch, I just mean that there is no difference in the choices or ending reached. Obviously there are a lot of places where the scenes seen along the way differ.

So first off, we have the S and HL counters prior to Lunch Time Evolution (LET). Two choices give you S+ or nothing, two give you HL+ or nothing, and one lets you choose HL+, S+, or nothing.

This gives the following count distribution

Code: Select all

(0, 0): 1/48 (2.0833%)
(0, 1): 1/16 (6.2500%)
(0, 2): 1/16 (6.2500%)
(0, 3): 1/48 (2.0833%)
(1, 0): 1/16 (6.2500%)
(1, 1): 1/6 (16.6667%)
(1, 2): 7/48 (14.5833%)
(1, 3): 1/24 (4.1667%)
(2, 0): 1/16 (6.2500%)
(2, 1): 7/48 (14.5833%)
(2, 2): 5/48 (10.4167%)
(2, 3): 1/48 (2.0833%)
(3, 0): 1/48 (2.0833%)
(3, 1): 1/24 (4.1667%)
(3, 2): 1/48 (2.0833%)

The cutoff is 2, so that gives the following distribution of actual choices seen at LET

Code: Select all

(False, False): 5/16 (31.2500%)
(False, True):  13/48 (27.0833%)
(True, False):  13/48 (27.0833%)
(True, True):   7/48 (14.5833%)

Of course this is symmetrical, so I'll just take the individual probability. The probability of being on a given path after LET is
0*5/16 + 1/2*13/48 + 1/3*7/48 = 53/288 (18.4028%)
The probability of being on neither Shizune nor Hanako/Lilly's path at this point is 91/144 (63.1944%). I'll abbreviate this path as X (compared to S and HL) to distinguish it from the case of running with Emi later.

Now, the next choice is whether to be serious with the nurse. However, this is completely independent of Cold War, so I'll handle Cold War next. The odds of choosing correctly are 1/2, so we have
S = HL = 53/576 (9.2014%)
X = 235/288 (81.5972%)

Now we have the nurse/run choices. Which gives 25% Emi, 75% not Emi. This is completely independent of the previous.


Now we have a few cases to consider

S & !E - 53/768 (6.9010%)
In this case, there's the kick out of room choice, so it is 50% K, 50% S
Kenji: 53/1536 (3.4505%)
Shizune: 53/1536 (3.4505%)

S & E - 53/2304 (2.3003%)
Here we have the funfunfun choice followed by whether to eat lunch with Shichan or Emi. So 50% K, 25% S, 25% E. That brings us up to
Emi: 53/9216 (0.5751%)
Kenji: 53/1152 (4.6007%)
Shizune: 371/9216 (4.0256%)

!S & E - 523/2304 (22.6997%)
100% Emi, enough said
Emi: 715/3072 (23.2747%)
Kenji: 53/1152 (4.6007%)
Shizune: 371/9216 (4.0256%)

HL & !E - 53/768 (6.9010%)
Here we have the walk with Lilly followed by the choice of Hanako or Lilly. So 50% K, 25% L, 25% H
Emi: 715/3072 (23.2747%)
Hanako: 53/3072 (1.7253%)
Kenji: 371/4608 (8.0512%)
Lilly: 53/3072 (1.7253%)
Shizune: 371/9216 (4.0256%)

X & !E - 235/384 (61.1979%)
Here we have the walk with Lilly followed by the art club. 75% K, 25% R
So the final tally is

Emi: 715/3072 (23.2747%)
Hanako: 53/3072 (1.7253%)
Kenji: 1243/2304 (53.9497%)
Lilly: 53/3072 (1.7253%)
Rin: 235/1536 (15.2995%)
Shizune: 371/9216 (4.0256%)

TL;DR; You're right. My earlier calculations were wrong.

[Mirage_GSM]

TL;DR; You're right. My earlier calculations were wrong.

Well, so were my earlier calculations, so I guess we're even. Thanks for making me recheck them.

[Atario]

I'm unreasonably fascinated by this.

Anyway, it is very true that the actual outcomes people have are far from random. Emi was my last route, because (a) I'm a lazy-ass and don't like to exercise, so I always told the nurse "maybe" and (b) I'm genre-savvy enough to know that choosing exercise is going to get you closer to the athletic girl, and I specifically held her off for last because I figured she would be a nice girl-next-door note to end on.

However, the devs' love of subverting genre meant that by trying to be nice to everyone, I insured my own Kenji ending those first two runs…

[Ryusashi]

..... wow..... so I had a 98% chance not to get hanko (the route I was most interested in when I read the char bios), and I got her on first shot...... I was fiddling around and had to look up how to get kenjis so I really am a statistical oddity in everything.

[Notguest]

I was fiddling around and had to look up how to get kenjis so I really am a statistical oddity in everything.

That's the one thing people say that I don't understand. It's very easy to find Kenji, even by just picking choices haphazardly.

[brythain]

I think the random statistical probability thing is just a mathematical curiosity. The chances that someone would play this game completely randomly are close to nil, which makes the premise invalid. What I prefer to think is that the choices mirror somewhat the internal process of reflection and/or cogitation that people go through when playing such games—that is, you get what you deserve. Or more seriously, you tend to head towards the girl whose attainment requirements most closely mirror what you're willing to be to an imaginary woman.

What would be really, really interesting would be a large sample of what paths people ended up with 1st time, and maybe 2nd or 3rd time. I ended up with Shizune first (good end) and then Kenji, then Lilly (good end), then Emi (both ends, ha) then Hanako (bad end, then good end), then Rin (good end, neutral end). I had to dig around a bit to get the other endings, but the Shizune bad end was an obvious one to find (binary choice, right?), followed by Hanako (indifferent, sad end) and then Rin. The hardest for me to find, surprisingly, was the Lilly neutral end, but it became a bit more obvious when I just chose whatever I hadn't chosen the first time round.

So if there are no really random choices, then the probabilities boil down to taking the sample of people likely to play KS (those who claim to have already done so) and figuring out what they were most likely to get as a specific population.

[Notguest]

Well in Act 1, I ended up with Kenji first, since I was trying to go for Lilly and didn't realize that requires chasing Hanako instead. There's a thread here from years back with a poll asking people what the first path they got was.

As for the completed game, I managed to get the good ending on the first try for every girl except Rin.

I'm a bit puzzled about why you would replay Act 1 every single time though. It doesn't even affect anything except a couple unimportant lines in Shinzune's route.

[brythain]

Well in Act 1, I ended up with Kenji first, since I was trying to go for Lilly and didn't realize that requires chasing Hanako instead. There's a thread here from years back with a poll asking people what the first path they got was.

As for the completed game, I managed to get the good ending on the first try for every girl except Rin.

I'm a bit puzzled about why you would replay Act 1 every single time though. It doesn't even affect anything except a couple unimportant lines in Shizune's route.

I think it's like this. I replayed Act 1 (with some skipping, of course) to get a feel for how the different routes made Act 1 look different in retrospect. Also, I replayed Act 1 because there were a lot of decision points that led to the other routes. And lastly, I replayed Act 1 for research purposes.

[Mirage_GSM]

Holy balls. More than a 50% chance of manly picnic. Makes me wonder how I didn't get it my first time.

..... wow..... so I had a 98% chance not to get hanko (the route I was most interested in when I read the char bios), and I got her on first shot...... I was fiddling around and had to look up how to get kenjis so I really am a statistical oddity in everything.

Please keep in mind that few - if any - readers picked totally random choices.
For example if you don't tell Kenji that Hanako is "not cute" - which no sane reader who is trying for her path would do - you greatly increase the chance of getting her ending. Not pissing off Lilly during Cold War and going to the library instead of to town in Sip (Part 2) are also no-brainers.
As for Kenji, there are four choices that trigger his ending, and each of them does so irrevocably. That said you only ever get to see one of those choices except for Emi's path (where you see none) and Rin's (where you see two).
By far the most people will have been sent off the roof by the choice in Mind your Step.
Even if you presume complete randomness Kenji's 53,95% are split as follows:
Mind Your Step: 34,05%
Creative Pain: 15,30%
No Recovery: 3,45%
Slow Recovery: 1,15%
This is because the overwhelmingly large probability of getting to the E/R route means you'll hardly ever see the choices in No Recovery or Slow Recovery.
Also it is also a very sneaky choice since it's not immediately obvious that not immediately disclosing your condition will lead to a bad end. As a roleplayer I asked myself "Would Hisao do that?" and considering his interior monologue up to this point he definitely wouldn't.

However, the devs' love of subverting genre meant that by trying to be nice to everyone, I insured my own Kenji ending those first two runs…

Uh, no.
If you try to be nice to everyone, you'll probably not get Shizune, Lilly or Hanako - because those require you to take their sides consistently, but Kenji's end is solely triggered by the four choices mentioned above. If you avoid those you'll most likely end up with Emi or Rin.

[Ryusashi]

..... wow..... so I had a 98% chance not to get hanko (the route I was most interested in when I read the char bios), and I got her on first shot...... I was fiddling around and had to look up how to get kenjis so I really am a statistical oddity in everything.

Please keep in mind that few - if any - readers picked totally random choices.
For example if you don't tell Kenji that Hanako is "not cute" - which no sane reader who is trying for her path would do - you greatly increase the chance of getting her ending. Not pissing off Lilly during Cold War and going to the library instead of to town in Sip (Part 2) are also no-brainers.
As for Kenji, there are four choices that trigger his ending, and each of them does so irrevocably. That said you only ever get to see one of those choices except for Emi's path (where you see none) and Rin's (where you see two).
By far the most people will have been sent off the roof by the choice in Mind your Step.
Even if you presume complete randomness Kenji's 53,95% are split as follows:
Mind Your Step: 34,05%
Creative Pain: 15,30%
No Recovery: 3,45%
Slow Recovery: 1,15%
This is because the overwhelmingly large probability of getting to the E/R route means you'll hardly ever see the choices in No Recovery or Slow Recovery.
Also it is also a very sneaky choice since it's not immediately obvious that not immediately disclosing your condition will lead to a bad end. As a roleplayer I asked myself "Would Hisao do that?" and considering his interior monologue up to this point he definitely wouldn't.

allow me to expand and explain. I figured id guess on hanakos. And ones i couldnt guess at, i flipped a coin. Landed her good ending (as I didn't want to smother her and wanted hisao to get his head straight which, in fact, he didn't...).

[Notguest]

Also it is also a very sneaky choice since it's not immediately obvious that not immediately disclosing your condition will lead to a bad end. As a roleplayer I asked myself "Would Hisao do that?" and considering his interior monologue up to this point he definitely wouldn't.

It was much worse back in the demo where the choice was just "I'm fine"