On the gathering of stones

[qinweiqi]

TLDR: Statistics is cool, and if you're looking for smooth river stones, it probably isn't worth staying in a single area for more than 10 minutes without finding any.

I recently noticed a post that requested more functionality on survey to be able to determine what kinds of stones can be foraged there. (viewtopic.php?f=7&t=4660)

This got me thinking about those elusive "smooth river stones" because I've never found one. At least, not if they appear with an object name of 'smooth river stone', which is what I've assumed. I know that prior to this exercise I've easily foraged at least 500 stones since they were released, most of those in rooms adjacent to rivers. So, I figured I'd gather a little data. I collected a little over 3000 stones of various sizes from a single location.

Here is a summary of the data:
Failed to find a stone: 2,246
Found a small stone: 1,780
Found a medium stone: 804
Found a large stone: 428

Side note: Also, the roll results showed that the distribution of rolls is pretty uniform, which I think is something we have all hoped and assumed to be true.

Now, I started off thinking that smooth river stones must be pretty rare, because again, I've gathered a good number of stones and never seen one. So I started off with an assumption that foraging for stones near a river has about a probability of 1/1000 to get a smooth river stone. Then I gathered a little over 3,000 stones and found none. That suggests that the probability of finding a smooth river stone is 1/5000 or less OR, much more likely, smooth river stones are not to be found every place one can forage stones next to a river.

Let's take another look at the data then. First, lets look at a summary of each stone type gathered:
Small stone found (roll results)
Minimum: 31
1st quartile: 48
Median: 65
Mean: 65
3rd quartile: 82
Maximum: 100

Medium stone found (roll results)
Minimum: 51
1st quartile: 64
Median: 74
Mean: 75
3rd quartile: 87
Maximum: 100

Large stone found (roll results)
Minimum: 61
1st quartile: 70
Median: 81
Mean: 80
3rd quartile: 90
Maximum: 100

You may have noticed something in those stats, I know I did when I first looked at them. In order to find a small stone, one must roll a 31 or higher, for a medium stone 51 or higher, or for a large stone 61 or higher. That gave me an idea to look at the data another way. This is what I found:

Rolls of 31+:
1,780 small stones
804 medium stones
428 large stones
637 failures

Rolls of 51+:
1,257 small stones
804 medium stones
428 large stones
106 failures

Rolls of 61+:
989 small stones
651 medium stones
428 large stones
0 failures

This brings out two important things. First, no roll of 61 or better failed to find a stone and second there were rolls between 31 and 60 that did fail to find stones. From these two observations, I conclude that the resource is determined first (small stone, medium stone, large stone) and then the check is made. Also, from the above, I concluded that it is extremely unlikely that smooth river stones may be found in the targeted foraging room. The reason we can be fairly certain that no smooth river stones are here to be found is that I still had over 2000 rolls above 60 that would suggest a rarity beyond 1/1000. I don't think smooth river stones are intended to be that insanely rare, otherwise people could spend hours to find just one that they could lose in seconds with a sling.

So, what is the probability of finding a smooth river stone? Can we infer anything from this data set where none were present to be found? We probably can. From the 61+ roll set, we can conclude that the stones are not uniformly distributed. It appears that the ratios are probably 50% small, 30% medium, 20% large (for this room, it may be different elsewhere). So, that means in a room where smooth river stones exist, I would expect that they would make up roughly 1% to 10% of the available stones. So our revised sets could be seen as 45% small, 27% medium, 18% large, 10% smooth river stone or 49.5% small, 29.7% medium, 19.8% large, 1% smooth river stone. Also, we can venture a guess at how high we'd have to roll to get one. Natural conclusions for me were 71+, 91+ or 100. These assumptions give us a range of probabilities for finding these as follows:
10% 71+ gives 0.03 => 75% probability to find at least one in 4 minutes
10% 91+ gives 0.01 => 75% probability to find at least one in 12 minutes
10% 100 gives 0.001 => 75% probability to find at least one in 2 hours
1% 71+ gives 0.003 => 75% probability to find at least one in 40 minutes
1% 91+ gives 0.001 => 75% probability to find at least one in 2 hours
1% 100 gives 0.0001 => so rare it's a cruel joke (75% for 1+ in 20 hours)

But, we should be able to do even better. We should be able to deduce that smooth river stones are present if we fail a roll on a 61+. This means that if we're watching closely (and it is very difficult to do so), our probabilities become much better:
10% 61+ => 3 minutes (pretty much guaranteed within 20 minutes)
5% 61+ => 6 minutes
1% 61+ => 30 minutes

So, if you forage for stones for 10 minutes and you either (1) get a smooth river stone or (2) fail on a 61+ roll, you can bet that smooth river stones are present. After that, there is no need to waste your time, move on and check someplace else.

Side note: If one looks at total stones gathered, the ratio is closer to 60%/26%/14%, which we would expect from the independent probabilities. Some of you might look askance as I did when I thought, 'hey, large stones are bigger - why are they harder to find?' I reason thusly, you're not looking for just any stone, you're looking for stones that are mostly balanced and mostly regularly shaped. The larger stones would be both less plentiful (recall 50/30/20 split above) and most likely stones of a larger size would tend toward being less balanced or more oddly shaped, further increasing the rarity.

Lastly, but not without significance, if you've stuck with me this far, you may be tempted to apply your new found wisdom and metagame (change your IC behavior due to new OOC knowledge). What I would suggest you do instead, is behave more IC -- if you were combing a river shore for nicely balanced chuckable stones would you pace back and forth over the same stretch of river picking over the stones? Maybe, but if you were not afraid to get lost and knew you didn't have to go back the way you came to get home, it's likely that you would just follow the river and pick for stones in a single direction. IE, you would have been doing it right to begin with. Happy stone foraging!

[Rias]

Riverstones can only be found in specific places, and aren't difficult to get if you know where they are. There's still a chance you'll get non-riverstones in these areas certainly, but I wouldn't call riverstones "rare" once you're in a place that produces them.

Keep up the exploring!