Surprising Seeds. Can Someone Help Math the REAL Chance?

[JazzyGames]

Halp plz.

So I know that when you read a message bottle after Day 150 a sunken chest spawns and populates with various loot. I have also come to understand that 60% is not the true effective chance of a surprising seed rolling.

The reason is because that roll occurs after the other rolls for loot have occurred, with normal loot rolling at varying amounts depending on the determined preset of the chest loot (of which there are five presets that appear to have different chances of rolling). If these other rolls result in a full sunken chest then the remaining rolls for items - including surprising seeds and trinkets - gets skipped. My sense from all of this is that the effective spawn chance of surprising seeds is significantly lower than the advertised 60%.

The process of rolling loot in messagebottletreasures.lua seems rather complex and I would love any insight from someone who is more familiar with this code than me. This would help temper expectations from sunken chest hunting and might also influence suggestions for possibly improving the chances of getting tree seeds from these chests. Out of the last 25 sunken chests I've cracked I have only seen around 7-8 surprising seeds. I am personally of the mind that it should be much higher if we are to go through the trouble of fighting Crab King for message bottles.

[Survivalist83]

All of my information comes from the wiki (Sunken Chest) and assumes that it is accurate. I have not looked at the code. With that in mind:

There are 9 slots in a sunken chest, and there is a 60% chance of a surprising seed spawning if there is at least one empty slot. So, we need there to be 8 or less items in the chest AND to hit the 60% chance. Since the 60% chance is known, all we need now is the chance of there being 8 or less items.

Given that there are 5 different, so we can calculate the chance of there being 8 or less items on each individually:

• The first preset has a 30% chance of occurring, and can spawn up to 6 items, so it always has an empty slot.
• The second preset has a 30% chance of occurring, and spawns a guaranteed 7 items, a 2/3 chance for another item, and a 50% chance for another item. This means that there is a 2/3 chance of it having an empty slot.
• The third preset has a 20% chance of occurring, and can spawn up to 7 items, so it always has an empty slot.
• The fourth preset has a 10% chance of occurring, and spawns a guaranteed 8 items with a 75% chance of spawning another item. This means that there is a 25% chance of it having an empty slot.
• The fifth preset has a 10% chance of occurring, and can spawn up to 8 items, so it always has an empty slot.

Multiplying these chances, 0.3 * 1.0 + 0.3 * 2/3 + 0.2 * 1.0 + 0.1 * 0.25 + 0.1 * 1.0 = 82.5% chance of that there will be an empty slot. 82.5% times the 60% of a seed spawning in the first place gives the answer: there is a 49.5% chance of a sunken chest having a seed after day 150.

You got 7-8 seeds (I'll call it 7.5) from 25 chests, meaning that you got a seed from 30% of chests, which is considerably less than 49.5%. I don't know if the difference is you being unlucky, an error in the translation between the code and the wiki, or a mistake on my part, but this is my understanding of it. I'm curious to see if someone else who better understands potential nuance in the code could shed some light on this.

tl;dr: assuming that the information on the wiki is accurate, you have a 49.5% chance of getting a surprising seed from a sunken chest, assuming the chest spawned after day 150.

[CremeLover]

1 hour ago, JazzyGames said:

Halp plz.

So I know that when you read a message bottle after Day 150 a sunken chest spawns and populates with various loot. I have also come to understand that 60% is not the true effective chance of a surprising seed rolling.

The reason is because that roll occurs after the other rolls for loot have occurred, with normal loot rolling at varying amounts depending on the determined preset of the chest loot (of which there are five presets that appear to have different chances of rolling). If these other rolls result in a full sunken chest then the remaining rolls for items - including surprising chests and trinkets - gets skipped. My sense from all of this is that the effective spawn chance of surprising seeds is significantly lower than the advertised 60%.

The process of rolling loot in messagebottletreasures.lua seems rather complex and I would love any insight from someone who is more familiar with this code than me. This would help temper expectations from sunken chest hunting and might also influence suggestions for possibly improving the chances of getting tree seeds from these chests. Out of the last 25 sunken chests I've cracked I have only seen around 7-8 surprising seeds. I am personally of the mind that it should be much higher if we are to go through the trouble of fighting Crab King for message bottles.

 

So, after dwelving in the code a bit, this is what I get:

There are 5 types of different chests presets, each has 0.x change of appearing, and each contain its own loot preset

saltminer	3
miner   	2
splunker	1
traveler	1
fisher   	3

The section where it adds an ancient seed is the following

if _container ~= nil and not _container:IsFull() then
	if math.random() <= math.clamp(
				TheWorld.state.cycles * TUNING.ANCIENT_TREE_SEED_CHANCE_RATE, TUNING.ANCIENT_TREE_SEED_MIN_CHANCE, TUNING.ANCIENT_TREE_SEED_MAX_CHANCE
	) then
		_container:GiveItem(SpawnPrefab("ancienttree_seed"))
	end
		if math.random() < TRINKET_CHANCE and not _container:IsFull() then
		local trinket = SpawnPrefab(trinkets[math.random(#trinkets)])
			_container:GiveItem(trinket)
	end
end

But this takes place after it has added the contents of the chests, and is executed regardless of what preset was selected, and it adds the seed inside the chest regardless. So unless I'm understanding something wrong, it should indeed be 60% chance after day 150.

But just to double check that _container:IsFull() wasn't doing anything strange, I tested using the following command for a sunken chest inst:components.container:NumItems(), which confirms that sunken chest have a max of 9 slots.

I tried spawning a chest with an ancient seed on it, but it turns out

The chest contents are not actually in the chest, but in a LIMBO inventory, which is still considered the chest inventory.

After doing some more testing with a few more chests

It seems sunken chest can indeed only contain 9 slots.
So if I understand this correctly...
If the chest doesn't have 9 items already, you get a 60% chance of having a seed.

Looking at the loot tables, only fisher and traveler types can contain 9 items.

Fisher has 2 items with a chance to be 0

oceanfishinglure_hermit_heavy = {0, 2},
scrapbook_page = {0,1},

 

Whereas traveler only has 1

scrapbook_page = {0,3},

 

I'm not going to do the probability, but my guess is that you have a guaranteed 60% chance on 60% of your rolls, and the chance decreases to 25%*60%=0.15% on Traveler, and 33.3%*50%*0.6%=0.1% on Fisher

[Survivalist83]

3 minutes ago, CremeLover said:

 

So, after dwelving in the code a bit, this is what I get:

There are 5 types of different chests presets, each has 0.x change of appearing, and each contain its own loot preset

saltminer	3
miner   	2
splunker	1
traveler	1
fisher   	3

The section where it adds an ancient seed is the following

if _container ~= nil and not _container:IsFull() then
	if math.random() <= math.clamp(
				TheWorld.state.cycles * TUNING.ANCIENT_TREE_SEED_CHANCE_RATE, TUNING.ANCIENT_TREE_SEED_MIN_CHANCE, TUNING.ANCIENT_TREE_SEED_MAX_CHANCE
	) then
		_container:GiveItem(SpawnPrefab("ancienttree_seed"))
	end
		if math.random() < TRINKET_CHANCE and not _container:IsFull() then
		local trinket = SpawnPrefab(trinkets[math.random(#trinkets)])
			_container:GiveItem(trinket)
	end
end

But this takes place after it has added the contents of the chests, and is executed regardless of what preset was selected, and it adds the seed inside the chest regardless. So unless I'm understanding something wrong, it should indeed be 60% chance after day 150.

But just to double check that _container:IsFull() wasn't doing anything strange, I tested using the following command for a sunken chest inst:components.container:NumItems(), which confirms that sunken chest have a max of 9 slots.

I tried spawning a chest with an ancient seed on it, but it turns out

The chest contents are not actually in the chest, but in a LIMBO inventory, which is still considered the chest inventory.

After doing some more testing with a few more chests

It seems sunken chest can indeed only contain 9 slots.
So if I understand this correctly...
If the chest doesn't have 9 items already, you get a 60% chance of having a seed.

Looking at the loot tables, only fisher and traveler types can contain 9 items.

Fisher has 2 items with a chance to be 0

oceanfishinglure_hermit_heavy = {0, 2},
scrapbook_page = {0,1},

 

Whereas traveler only has 1

scrapbook_page = {0,3},

 

I'm not going to do the probability, but my guess is that you have a guaranteed 60% chance on 60% of your rolls, and the chance decreases to 25%*60%=0.15% on Traveler, and 33.3%*50%*0.6%=0.1% on Fisher

If I'm understanding this correctly, I believe that the wiki was accurate, so the 49.5% I found should be correct. I find it weird that a chest can only have 9 slots, I wonder why that is.

Thank you for taking the time to dive into the code!

[CremeLover]

2 minutes ago, Survivalist83 said:

If I'm understanding this correctly, I believe that the wiki was accurate, so the 49.5% I found should be correct. I find it weird that a chest can only have 9 slots, I wonder why that is.

Thank you for taking the time to dive into the code!

Correct! Maybe I should've looked at there first.

Thanks for correctly doing the numbers, I forgot how probability works

[JazzyGames]

Wow that’s a major difference in the expected outcome. You both are amazing, thank you so much.

[CremeLover]

24 minutes ago, Survivalist83 said:

Multiplying these chances, 0.3 * 1.0 + 0.3 * 2/3 + 0.2 * 1.0 + 0.1 * 0.25 + 0.1 * 1.0 = 82.5% chance of that there will be an empty slot. 82.5% times the 60% of a seed spawning in the first place gives the answer: there is a 49.5% chance of a sunken chest having a seed after day 150.

Wait, after looking at this again...

This is your equation:

0.3 * 1.0 + 0.3 * 2/3 + 0.2 * 1.0 + 0.1 * 0.25 + 0.1 * 1.0

Or

0.3 * 1.0
0.3 * 2/3
0.2 * 1.0
0.1 * 0.25
0.1 * 1.0

But I've realized there's something missing

26 minutes ago, Survivalist83 said:

The second preset has a 30% chance of occurring, and spawns a guaranteed 7 items, a 2/3 chance for another item, and a 50% chance for another item. 

And so:

0.3 * 1.0
0.3 * 2/3 * 0.5
0.2 * 1.0
0.1 * 0.25
0.1 * 1.0

0.3 * 1.0 + 0.3 * 2/3 * 0.5 + 0.2 * 1.0 + 0.1 * 0.25 + 0.1 * 1.0=72.5%

So, by adjusting the numbers, I get the chance to be be 43,5% instead

[lowercase skye]

18 minutes ago, Survivalist83 said:

I find it weird that a chest can only have 9 slots, I wonder why that is.

There's unused animations for Sunken Chests being unlocked and opened like a regular chest, which can actually be seen happening in the animated short for She Sells Seashells, which implies that you were once supposed to be able to open up the chests and access their inventory in that way. I imagine 9 slots was chosen to balance them in an environment where you could unlock and use them as storage, and it was simply never adjusted once the mechanic was removed.

[Survivalist83]

3 minutes ago, CremeLover said:

0.3 * 2/3 * 0.5

It's a guaranteed 7 items, plus a 2/3 chance for another, plus a 1/2 chance for another. We can ignore the 7 since it's guaranteed; we basically only care if the 2/3 chance fails and/or the 1/2 chance fails, because if at least one fails, there will be an empty spot for the seed. In other words, as long as we don't get both, we can get a seed.

The chance for getting both is 2/3 * 1/2 = 1/3. So, the chance of not getting both is 1 - 1/3 = 2/3. I think the confusion came from 2/3 being both the final answer and the chance at getting the first item.

I'm pretty sure this is right, so it still should be 49.5%. That being said, statistics are tricky so thank you for double-checking - if anything seems wrong with this response, please let me know

[CremeLover]

7 minutes ago, Survivalist83 said:

It's a guaranteed 7 items, plus a 2/3 chance for another, plus a 1/2 chance for another. We can ignore the 7 since it's guaranteed; we basically only care if the 2/3 chance fails and/or the 1/2 chance fails, because if at least one fails, there will be an empty spot for the seed. In other words, as long as we don't get both, we can get a seed.

The chance for getting both is 2/3 * 1/2 = 1/3. So, the chance of not getting both is 1 - 1/3 = 2/3. I think the confusion came from 2/3 being both the final answer and the chance at getting the first item.

I'm pretty sure this is right, so it still should be 49.5%. That being said, statistics are tricky so thank you for double-checking - if anything seems wrong with this response, please let me know

Ah, well, guess I do indeed need to refresh on my statistics theory then, or just have a nice night of sleep, or both ^-^'

Thanks for double checking the double check!

[Survivalist83]

I figured I'd also calculate expected number of surprising seeds you get by killing crab king, assuming it is past day 150 and you've returned the cracked pearl. I'll put a tl;dr at the end for those who aren't as interested in the math behind it.

CK drops 2 guaranteed bottles, a third with a 50% chance and a fourth with a 25% chance. This leads to these chances of getting x bottles:

• 2 bottles: (1 - .5) * (1 - .25) = 37.5%
• 3 bottles: (.5) * (1 - .25) + (1 - .5) * (.25) = 50% (note that you could get 3 bottles by getting either the third but not the fourth, or the fourth but not the third)
• 4 bottles: (.5) * (.25) = 12.5%

Then, given that 49.5% of chests contain surprising seeds, we can calculate the expected number of surprising seeds for each of the three options above (note that I'm rounding to one decimal since we don't need eons of precision)

• Assuming you get 2 bottles:
  • 0 seeds: 1 * (.495)^0 * (.505)^2 = 25.5%
  • 1 seed: 2 * (.495)^1 * (.505)^1 = 50.0%
  • 2 seeds: 1 * (.495)^2 * (.505)^0 = 24.5%%
• Assuming you get 3 bottles:
  • 0 seeds: 1 * (.495)^0 * (.505)^3 = 12.9%
  • 1 seed: 3 * (.495)^1 * (.505)^2 = 37.9%
  • 2 seeds: 3 * (.495)^2 * (.505)^1 = 37.1%
  • 3 seeds: 1 * (.495)^3 * (.505)^0 = 12.1%
• Assuming you get 4 bottles:
  • 0 seeds: 1 * (.495)^0 * (.505)^4 = 6.5%
  • 1 seed: 4 * (.495)^1 * (.505)^3 = 25.5%
  • 2 seeds: 6 * (.495)^2 * (.505)^2 = 37.5%
  • 3 seeds: 4 * (.495)^3 * (.505)^1 = 24.5%
  • 4 seeds: 1 * (.495)^4 * (.505)^0 = 6.0%

Bear with me here, we're almost done. Now, we can add the results of each possibility occurring:

• 0 seeds: 37.5% * 25.5% + 50% * 12.9% + 12.5% * 6.5% = 16.83% (note: it is in fact NOT 24.13% like I originally thought, I miscalculated it. 16.83% is the correct number)
• 1 seed: 37.5% * 50.0% + 50% * 37.9% + 12.5% * 25.5% = 40.89%
• 2 seeds: 37.5% * 24.5% + 50% * 37.1% + 12.5% * 37.5% = 32.43%
• 3 seeds: 50% * 12.1% + 12.5% * 24.5% = 9.11%
• 4 seeds: 12.5% * 6.0% = 0.075%

tl;dr: Assuming you're past day 150 and have returned the cracked pearl, you have a 16.83% chance of getting 0 seeds from CK, a 40.89% chance to get 1 seed, a 32.43% chance to get 2 seeds, a 9.11% chance to get 3 seeds, and a 0.075% chance to get 4 seeds. This means that there is a 73% chance of getting either 1 or 2 seeds.

One final note: I believe that all of this is correct, but I'm only human and wouldn't mind if someone good at math would double-check my work to make sure I didn't make a mistake somewhere. (edit: I caught one error already, the chance at 0 seeds is 16.83%, not 24.13%. This is proof I'm only human lol)

[Wumpair]

5 hours ago, Survivalist83 said:

I figured I'd also calculate expected number of surprising seeds you get by killing crab king, assuming it is past day 150 and you've returned the cracked pearl. I'll put a tl;dr at the end for those who aren't as interested in the math behind it.

CK drops 2 guaranteed bottles, a third with a 50% chance and a fourth with a 25% chance. This leads to these chances of getting x bottles:

• 2 bottles: (1 - .5) * (1 - .25) = 37.5%
• 3 bottles: (.5) * (1 - .25) + (1 - .5) * (.25) = 50% (note that you could get 3 bottles by getting either the third but not the fourth, or the fourth but not the third)
• 4 bottles: (.5) * (.25) = 12.5%

Then, given that 49.5% of chests contain surprising seeds, we can calculate the expected number of surprising seeds for each of the three options above (note that I'm rounding to one decimal since we don't need eons of precision)

• Assuming you get 2 bottles:
  • 0 seeds: 1 * (.495)^0 * (.505)^2 = 25.5%
  • 1 seed: 2 * (.495)^1 * (.505)^1 = 50.0%
  • 2 seeds: 1 * (.495)^2 * (.505)^0 = 24.5%%
• Assuming you get 3 bottles:
  • 0 seeds: 1 * (.495)^0 * (.505)^3 = 12.9%
  • 1 seed: 3 * (.495)^1 * (.505)^2 = 37.9%
  • 2 seeds: 3 * (.495)^2 * (.505)^1 = 37.1%
  • 3 seeds: 1 * (.495)^3 * (.505)^0 = 12.1%
• Assuming you get 4 bottles:
  • 0 seeds: 1 * (.495)^0 * (.505)^4 = 6.5%
  • 1 seed: 4 * (.495)^1 * (.505)^3 = 25.5%
  • 2 seeds: 6 * (.495)^2 * (.505)^2 = 37.5%
  • 3 seeds: 4 * (.495)^3 * (.505)^1 = 24.5%
  • 4 seeds: 1 * (.495)^4 * (.505)^0 = 6.0%

Bear with me here, we're almost done. Now, we can add the results of each possibility occurring:

• 0 seeds: 37.5% * 25.5% + 50% * 12.9% + 12.5% * 6.5% = 24.13%
• 1 seed: 37.5% * 50.0% + 50% * 37.9% + 12.5% * 25.5% = 40.89%
• 2 seeds: 37.5% * 24.5% + 50% * 37.1% + 12.5% * 37.5% = 32.43%
• 3 seeds: 50% * 12.1% + 12.5% * 24.5% = 9.11%
• 4 seeds: 12.5% * 6.0% = 0.075%

tl;dr: Assuming you're past day 150 and have returned the cracked pearl, you have a 24.13% chance of getting 0 seeds from CK, a 40.89% chance to get 1 seed, a 32.43% chance to get 2 seeds, a 9.11% chance to get 3 seeds, and a 0.075% chance to get 4 seeds. This means that there is a 73% chance of getting either 1 or 2 seeds.

One final note: I believe that all of this is correct, but I'm only human and wouldn't mind if someone good at math would double-check my work to make sure I didn't make a mistake somewhere.

This seems like a fun discussion, so let me do a comment about probability.

In general your calculations look corect, but there is one detail that I am not sure how it works in the code and maybe is worth to mention.

The chances of getting 2/3/4 bottles can be interpreted in two ways.

So, lets imagine that there are 4 possible bottles you can get: B1, B2, B3, B4. B1 and B2 drop 100% of the time; but now there are two ways to interpret the drop rate of B3 and B4

Option 1: Both bottles are independent from each other, and B3 has a 50% chance to drop, while B4 has a 25% chance to drop. Then what you state for the probabilities to get 2/3/4 bottles is correct, it is 37.5%, 50% and 12.5%; for instance to get 3 bottles you can get it by either getting (B1, B2, B3) or (B1, B2, B4) = 0.5×0.75 + 0.5×0.25 = 0.5

Option 2: B3 still has a 50% drop rate, but only if we get B3 then there is a 25% to also get B4. So this would be a conditional probability, this is the case if there is an ordering of the bottles, that you can only get 4 bottles if you rolled for 3 bottles before. In this case the probabilities shift. Now the probability of getting 2 bottles is 50% (You didn't get B3, and so also didn't get B4), getting 3 bottles is 37.5% (You got B3 but not B4 = 0.5×0.75) and for 4 bottles 12.5% (0.5×0.25)! So your probabilities for getting 2 and 3 bottles switch, and as such the expected average also becomes smaller.

The question is, how is it written in the code (I have no idea so maybe someone could clarify).

But yeah, I just wanted to mention - be careful when you work with probabilities, because sometimes you can have multiple interpretations of the same sentance :]

And even further, maybe you could interpret the numbers in the wiki as "you have some chance to get 2 bottles, 50% to get 3 and 25% to get 4". Based on this, you could say that you have a 25% chance to get 2 bottles (100-50-25). This is probably wrong, but then again the 50% and 25% are ambiguos.

There is a famous problem in probability (Bertrand paradox), where depending on the method you aplly when asuming what "line segment" means, you can get 3 different probabilities. In this particular case, best way to check: Look at what the code says directly, or via simulation (kill ck a bunch of times and see how many times you get 2, 3 or 4 bottles) :P

[Survivalist83]

54 minutes ago, Wumpair said:

This seems like a fun discussion, so let me do a comment about probability.

In general your calculations look corect, but there is one detail that I am not sure how it works in the code and maybe is worth to mention.

The chances of getting 2/3/4 bottles can be interpreted in two ways.

So, lets imagine that there are 4 possible bottles you can get: B1, B2, B3, B4. B1 and B2 drop 100% of the time; but now there are two ways to interpret the drop rate of B3 and B4

Option 1: Both bottles are independent from each other, and B3 has a 50% chance to drop, while B4 has a 25% chance to drop. Then what you state for the probabilities to get 2/3/4 bottles is correct, it is 37.5%, 50% and 12.5%; for instance to get 3 bottles you can get it by either getting (B1, B2, B3) or (B1, B2, B4) = 0.5×0.75 + 0.5×0.25 = 0.5

Option 2: B3 still has a 50% drop rate, but only if we get B3 then there is a 25% to also get B4. So this would be a conditional probability, this is the case if there is an ordering of the bottles, that you can only get 4 bottles if you rolled for 3 bottles before. In this case the probabilities shift. Now the probability of getting 2 bottles is 50% (You didn't get B3, and so also didn't get B4), getting 3 bottles is 37.5% (You got B3 but not B4 = 0.5×0.75) and for 4 bottles 12.5% (0.5×0.25)! So your probabilities for getting 2 and 3 bottles switch, and as such the expected average also becomes smaller.

The question is, how is it written in the code (I have no idea so maybe someone could clarify).

But yeah, I just wanted to mention - be careful when you work with probabilities, because sometimes you can have multiple interpretations of the same sentance :]

And even further, maybe you could interpret the numbers in the wiki as "you have some chance to get 2 bottles, 50% to get 3 and 25% to get 4". Based on this, you could say that you have a 25% chance to get 2 bottles (100-50-25). This is probably wrong, but then again the 50% and 25% are ambiguos.

There is a famous problem in probability (Bertrand paradox), where depending on the method you aplly when asuming what "line segment" means, you can get 3 different probabilities. In this particular case, best way to check: Look at what the code says directly, or via simulation (kill ck a bunch of times and see how many times you get 2, 3 or 4 bottles) :P

This is a very good point. My understanding of DST and the DST wiki in general (although I don't know if I'm right) is that additional drops, like the fourth bottle in this case, are independent of each other. You correctly identified that I did the calculation with this in mind. Thank you for pointing this out the potential confusion here! I'll keep this in mind for the future.

6 hours ago, Survivalist83 said:

0 seeds: 37.5% * 25.5% + 50% * 12.9% + 12.5% * 6.5% = 16.83% (note: it is in fact NOT 24.13% like I originally thought, I miscalculated it. 16.83% is the correct number)

This is proof I'm only human and make mistakes. I found this one because I realized that the chances I gave added up to 106.63%, not 100% like they should - I redid the numbers and it turns out that the chance of getting 0 seeds is lower than I thought. I made an edit to my original post to fix it. If anyone sees anything else wrong with this, please let me know and I'll fix it.

As a side note, now they add up to 99.33%. This is due to compounded rounding errors, but since it's only 0.0067 off, I don't think it really matters. I'm fine with this because a lot of the numbers that aren't rounded have like 8+ decimals, and it seems pointless to me to have more than 2 decimals for an approximation like this. If for some reason you want more than 2 decimals of accuracy, I've shown my work and you can calculate for yourself to however many decimals you want.