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    Your Elvenar Team

Snow Flurry Probabilities (GP's and Bldgs)

DeletedUser2424

Guest
Background:
I felt like nerding out on some probability math. Since you can win snowflakes which in turn increases the number of possible attempts it makes for an interesting yet solvable math puzzle. In my quest for how to properly apply "binomial distribution models" to the problem... I ended up just writing a simple python script to simulate the probability of getting Grand Prize 2 & Grand Prize 3. Here's the results.

TL;DR:
- You have < 1% chance of getting the 3rd grand prize without diamonds.
- You will be able to get the 2nd grand prize (no luck needed) if you complete all quests.
- You will earn about 14 event buildings (if you complete all quests).
- If you buy small chests, you'll end up with more daily exclusive buildings then you would the large chests (assumes your trading in lots of snowflakes not a single chest) but you have a slightly lower chance of getting the grand prizes and will get less KP, etc.

Results
2016-12-20_15-34-24.png
More Detailed
Includes all 3 box options as well as the amount of runes, KP, and relics you will gain.
2016-12-20_17-20-18.png

So, which box is the best?

Well it turns out the choice isn't as obvious as we all once assumed... Large Box seems like the best option but for many of you I suspect the medium box or small box will be your best bet.

Large Box:
Pros: Most KP, and a practically negligible 1% better chance of getting a 3rd Grand Prize
Cons: Least Tier 3 Relics

Medium Box:
Pros: Most Rune Shards (~10 more at 6000)
Cons: Least KP (about 50 less than Large)

Small Box:
Pros: Most Daily Exclusive Buildings (about 2 more)
Cons: Least Rune Shards


More on test methodology:
- The test essentially simulates how the game works... you trade in 140 snowflakes and have 7% chance of getting 200 snowflakes, a 13% chance of getting 140 snowflakes, and 25% chance of getting a building. It uses a random number generator for each trade in and increases or decreases snowflakes accordingly until you can't play anymore. The simulation is run over 100,000 times to get a statistically accurate result.
- Only the big chest is used (140 flakes for 20 globes)
- Someone on SoggyShort's spoilers thread (which is awesome, sorry couldn't find the exact post) estimated of number of Snow Flakes you could earn (1156 daily quests / 4128 secondary quests / 1104 dropped snow crystals) giving you 6388 flakes without diamonds. I used those.
- 650 globes are needed for Grand Prize 2
- 1450 globes (in total) are needed for Grand Prize 3.
- Source Code for the python script: http://pastebin.com/amAREij0

Anticipated Questions...

> Something must be wrong with your math... There is only a 25% chance of getting a building...
Yes, you have a 25% of building for each chest opened. But! since you have 20% chance (13% + 7%) of getting a free chest by winning snowflakes. This improves the probability to at least 25/80 or 31.5% (it's actually even better since 7% of the time you get 1.42 chests). Doing a simple math of say i have 4000 snowflakes / (140 flakes / chest) * .25 is not taking all this into account.

> So am i better of buying snowflakes or a premium building?
While, i haven't thought this through completely since I don't plan on buying either. I would think of it like this... You'd need about 2000 snowflakes to comfortably assure you'll get the Grand Prize 3 (which is 8500 diamonds to buy). After reading this post... the Mario Patron Saint is "more efficient" than the 3rd Grand Prize (the tower) and at only 3000 a pop you could buy almost 3 of them. Which is most likely a much better spend. On the other hand, you'll stand to pickup an extra 4-5 daily exclusive buildings and other goodies in the process... but placing 18 3x3 buildings you'll likely be straining your coin supply anyway.

EDIT: 12/20 Added Large vs. Small Chest
EDIT2: 12/20 Added Medium Box and comparison of runes, shards, and relics gained.
 
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DeletedUser5800

Guest
I'll take the fact that you are guaranteed to win GP2 without diamonds!
Somebody will probably find a way to be so unlucky they still don't, I missed it in the Solstice by one roll I think, not sure what the odds on that were... I know this time I plan on being the one person that makes it to the third just so I can tell Soggy about it. :eek:
 

DeletedUser2424

Guest
Somebody will probably find a way to be so unlucky they still don't, :eek:

That person will be liar. ;) You only need 4550 snowflakes and even if you didn't pick up a single snowflake you could still get 5284 from just the quests.
 

DeletedUser2424

Guest
Can you run the test for the other chests as well? I'm curious how the lower snowglobe efficiency balances with the higher chance of snowflakes

I updated the my initial post with the results of the simulation. It's interesting... you end up with more daily exclusive rewards overall that way... but you'll have a worse chance of getting the Grand Prizes.
 

SoggyShorts

Well-Known Member
I updated the my initial post with the results of the simulation. It's interesting... you end up with more daily exclusive rewards overall that way... but you'll have a worse chance of getting the Grand Prizes.
VERY interesting.... If you ignore the >1% chance to get the third prize maybe it's better to go for small boxes...
I'm working on a version right now that will also include the medium box and also include the KP / RuneShards / Relics you'd gain.
I'm very interested in this as well, especially since I don't realy need any tier 3 relics, and wouldn't love getting them 30% of the time
 

DeletedUser2424

Guest
Added it. And once again... i'm stunned... Their is no clear best or worst box. Each has their own benefits...

Large Box:
Pros: Most KP
Cons: Least Relics

Medium Box:
Pros: Most Rune Shards (~10 more at 6000)
Cons: Least KP (about 50 less than Large)

Small Box:
Pros: Most Daily Exclusive Buildings (about 2 more)
Cons: Least Rune Shards
 
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DeletedUser5800

Guest
That's all pretty awesome stuff, I think I'll stick big since I like Kp's more than runes... and that .4 chance at the grand prize is the best chance I've had at anything in years. :D
 

SoggyShorts

Well-Known Member
Wow, I think I'm going to go all small boxes. For my cities
2 Daily exclusives>50 kp, and I don't care about runes or relics at this point. 2 Days of KP isn't nearly as hard to get as the rest.
 
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DeletedUser4329

Guest
Hey fasdf, loving your analysis so far.

Something that doesn't make sense to me is how the small boxes result in an increased chance of event buildings. Based on the probabilities the large boxes should give a better chance overall, even taking into account that you get more attempts (at a much lower probability) with small boxes.

Could it be that the particular run-through you did with your python script rolled the dice a certain way and a re-run would give a very different outcome?
 

SoggyShorts

Well-Known Member
Something that doesn't make sense to me is how the small boxes result in an increased chance of event buildings.
I think part of the reason is that when you win snowflakes from a small box you can win 90, which is almost 2 boxes, but in a big box you win 200 which isn't even a box and a half.
Edit, yeah that's got to be why, especially since you have a 30% chance of free snowflakes on a small box instead of 20% chance on a big.
 

DeletedUser2424

Guest
Could it be that the particular run-through you did with your python script rolled the dice a certain way and a re-run would give a very different outcome?

Math Answer:
The adjusted probabilities for each are 10% for small, and 31.5% for large. (See my Anticipated Question Section of initial post for where these come from).

But, since we get about 3 chances for every small chance the probability can be calculated by calculating the probability of you not getting a chest * 3 (since it's an indepent trial) and then subtracting the product from 1 to get the probability of getting a building.

90% * 90% * 90% = 0.729 (is the probability of not getting one)... You then invert that to get the probability of getting one of the buildings.
1 - 0.729 = 27.1%

So, ok... 27.1% is still lower than 31.5%...

But, those simple probabilities of 10% and 31.5% aren't completely accurate... it doesn't account for the fact that you get a better bonus in the small chest case. In the case of the large chest you get 200 which is 200/140 = 1.42 chests. But, in the case of the small chest... it's 90/50 = 1.8 chests!!! So that certainly tips the scales more towards the small chest and likely makes up that difference... Because we achieve more rolls easier in this case.

So... this is where simulation becomes super handy... and that's why i wrote the script because fully figuring it out with math hurts.


I'm certainly a flawed human answer:
So, the way my script works is runs through 100k simulations and then takes an average so. It effectively accounts for luck. I've run the script multiple times and got results differing by at most like 0.03.

That being said. The results here are only as good as the human that input the data. I'm pasting below the relevant parts of the script that would be the most susceptible to human error. I would love it if you could verify them independently.... See the last block for the small chest I added comments to make it make sense. I used a self.cost variable to handle the different cases small, medium, or large.

Code:
             randNum = random.random();   // Python random returns a uniform random distribution between 0 and 1.0

             # For large Chest          
             if self.cost == 140:
                if randNum < 0.1300:
                    self.money += 140
                elif randNum randNum < 0.2000:
                    self.money += 200
                elif randNum randNum < 0.4500:
                    self.buildings += 1
                elif randNum < 0.58:
                    self.kp += 25
                elif randNum < 0.70:
                    self.runeShards += 5
                else:
                    self.relics += 12

            # Mediuum Chest
            elif self.cost == 90:
                if randNum < 0.10:
                    self.money += 90
                elif randNum < 0.24:
                    self.money += 140
                elif randNum < 0.37:
                    self.buildings += 1
                elif randNum < 0.50:
                    self.kp += 10
                elif randNum < 0.64:
                    self.runeShards += 3
                else:
                    self.relics += 6

            # Small Chest
            randNum is between 0.00 and 1.00
            elif self.cost == 50:
                # if num 0 --> 0.13
                if randNum < 0.1300:
                    self.money += 50
                # If number is > 0.13 and < 0.30  = difference of .17
                elif randNum < 0.30:
                    self.money += 90
                # If number is between 0.30 and 0.37 - difference of 7%
                elif randNum < 0.3700:
                    self.buildings += 1
                # if number is between 0.37 and 0.51 - difference of 14%
                elif randNum < 0.51:
                   self.runeShards += 1  # Add 1 rune shard
                #if number is between 0.51 and 0.64 - difference of 13%
                elif randNum < 0.64:
                   self.kp += 5  # 5 kp's are rewarded
                # If > 0.64 and less than 1.0  (difference of 36%)
                else:
                   self.relics += 3

Here's the full source... it's a bit messy... the code started clean and got a bit hacky once I added all these different chest / relic options.
http://pastebin.com/VM55KMSd



@fasdfTTT: May include your probability computations in my spreadsheet?

Absolutely!
 

SoggyShorts

Well-Known Member
@fasdfTTT
Classic mistake. A few of us over on Beta tried approximating the odds, but didn't go nearly as far as you've taken it, and didn't get nearly as accurate a result. So what was your mistake? "If you do a job too well, you’ll get stuck with it."
I'll be counting on you for the Valentines/Easter/St .Patrick's......
Now, SoggyShorts? That guy knows what he's doing. Copy, Paste, Hero.:D

Seriously though, well done, and thank you!
 
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