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Old 01-28-2008, 12:58 AM   #1
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Lightbulb [ GUIDE ] Arte's Gem Hunt - EXPLAINED !!

Fact, or "Artefact" ?
Williamson_39

We see things that are not there in the Gem Hunt. Through a combination of causes, it is easy to be misled by past successes, and think that there is a specific reason you found what you did, where you did, or when you did. As a result, we are drawn to tips about which rocks to pick that are well-intentioned and seemingly logical, but unfortunately baseless. In this post, I will expose the facts and try to explain the "cognitive trap" that we all tend to fall into.

Here's the inside scoop.

Spoiler Alert: This information contains details that will simplify your experience and interaction with the mines, and may take away some of the fun for some. All you really need to know is #1 below. If you are interested in how it works, read beyond those, otherwise stick with the first fact and go play - other tips won't hurt you in this case, and if they're fun, go for it!


Facts.
  1. Gems are distributed among the five mines based on their colour. Justin's thread (sticky in this forum) will show you the layout, but basically each colour of gem has its "home" mine that happens to match the colour of gem. The lower level gems also appear in one or two other mines. If you do the math per mine (rather than by gem colour) you will find that this means that there are exactly 14 gems in every mine (6 of the mine's background colour gems, 2 "uncommon" gems of another colour, and 6 "common" gems of two other colours (3 each). This is the ONLY critical piece of information you need to know to get a crown of wonder.
    .
  2. Bats, spiders, sparkles, drips, and rock positioning are random and completely independent of gems and slag. Not one of these provides any clue as to where you will find a gem.
    .
  3. "Rare" gems are only rare because they are only found in their "home" mine. But in a given mine, the top gem has the same chance of being found as any of the other 13 available there.
    .
  4. Rocks are not really "hiding" places, they are just things to click. Gems are not placed and covered by rocks in the mine at all.


    Each time you pick a rock (any rock!), two things happen:
    • You have a 7 in 20 chance of finding some sort of gem (something that is not slag) on your first rock, 7 in 19 on the second, and 7 in 18 on the third, regardless of how many rocks are actually visible in the mine (they vary).
    • IF you find a gem, it will be one of the 14 associated with that mine, chosen randomly based on equal probabilities.
Since you have three tries, the probability of finding nothing but slag on a given day is only:

Click the image to open in full size.

or 25%. So conversly, three quarters of the time you play, you will find a gem. Those are pretty good odds for success! But exactly which gem you find is still random, so your chances of finding one particular gem are much lower. More on that later.

Equal probability among gems in a mine means that for example, in the Buried Bones Mine, you are as likely to find the Webkinz Diamond as the Booger Nugget. By contrast, over the entire map, the Booger Nugget is more common because it appears in three different mines. That's why you tend to fill your gem box with common gems first - because you find them "accidentally" in other mines and because there are 9 different common gems and only one rare in each mine. But their individual probabilities in a single mine are the same.

The clue to #4 above came from the realization that sometimes there are fewer rocks shown than there are gems available in that mine, and you still find slag in most single clicks, but the odds of success don't change. If there are fewer rocks than gems, and some are always "slag," that would mean that some gems could not be hidden! But people complete their collections relatively quickly, so I felt it was extremely unlikely that certain gems would ever be "unfindable."


"Artefacts"
So why do we see patterns in the sparkles and critters where there are none? Why does that rock over there seem to work all the time? It is the result of something referred to as "cognitive dissonance bias". We have a natural tendency to filter our perceptions to discard inputs that contradict our existing ideas, and in the process ignore failures and over-emphasize successes. We don't know we're doing this, but we love being right. It often requires overwhelming evidence to change our thinking.

A certain sparkle pattern catches our eye as "unique", and so we click those rocks. If we find a gem that day (which we will do almost 75% of the time), we think, AHA! That might be the trick! If we are unlucky enough to find another one the next time using the same steps (which we will do more than half the time!!), even if it was not exactly the same, we subconsciously ignore any "trivial" differences, our hypothesis is confirmed, and our superstition reinforced. If the next one fails, we frown and wonder what WE did wrong. But the next success reaffirms our faith, and we promptly forget about the evidence to the contrary. We were "right" three out of four days, so we MUST have solved the game! I know the trick to gem hunt!!!

I know I'm oversimplifying, but does that sound familiar?

This cognitive trap is extremely easy to fall into - all but inevitable really - because we are successful three out of every four days NO MATTER WHICH ROCKS WE PICK. That provides ample opportunity to "prove" just about any theory.


Completing your collection for the Crown of Wonder
When you first get started, choosing a mine is easy. But as your collection grows, you can use the distribution of gems to your advantage, so that when you choose a mine, you are giving yourself the best possible chance to find a gem you need. WI member ennazusjsc has a great tool to optimize your daily search, by prioritizing the mines based on the distribution of gems and your current collection. Based on the new probability information above, check out her revised version later in that thread too. It also incorporates a weighting for the potential value of duplicates that you can sell to Arte.

When you get down to only a few gems remaining, you must be patient! Based on the four Facts above, there is NO WAY to improve your chances of finding that last missing gem, other than being sure to look in a mine that holds it. Particularly for a top level "rare" gem, you MUST look in its home mine to ever find it.

Your chance of finding a particular gem that you need is a little bit complicated. The overall probability on a given day is the sum of the probabilities of each turn. But those turns are not the same because you don't always have three chances at it - your day ends if you find any other gem. In your first turn, it's simple - the probability of finding a particular gem is just the probability of NOT finding slag, times the probability of getting the one gem you need out of the 14 in that mine. But in order to even have a second turn, you need to have found slag on the first try, and similarly with turn three, the first two had to have been slag. So, the probability (P) of finding one specific gem on a given day is:

You find it on your first try: (P(not slag in turn 1) AND P(one gem)) OR
Second try: (P(slag in turn 1) AND P(not slag in turn 2) AND P(one gem)) OR
Third try: (P(slag in turn 1) AND P(slag in turn 2) AND P(not slag in turn 3) AND P(one gem))

Whew! In probability, "AND" means multiplication, and "OR" means addition. P(one gem) is always 1/14, or about 0.07. So we can plug in the numbers for turns 1 to 3 respectively, and add them up:

Click the image to open in full size.

If my math is correct (repaired, thanks jillion!), your chance of finding that last missing gem today, as long as you go into a mine that holds it, is about 1 in 20 (actually a little better than that - the answer is rounded). Incidentally, this is the same chance you have of finding the gem-of-the-day, but again, only if you look a mine that holds it. Otherwise, your chances are nil.

NB. Because the gems are equally weighted, there is a simpler way to arrive at the same answer: The probability of finding one gem on a given day is just the probability of finding any gem, 1 - P(slagDay) from the first equation above, divided by 14. This yields exactly the same answer in this case, but does not expose the different contribution of each turn, which might be useful to verify the theoretical probabilities against observed results. On the other hand, it might just be needlessly baffling.

1 in 20 (5%) seems like a slim chance, but it's actually pretty good news. It means that 50% of the time, you will find your last missing gem in two weeks or less. Nine times out of ten, you'll be stuck with one gem remaining to finish your collection for less than a month and a half.

Unfortunately, as jillion points out below, there will be times when you are that one in ten that takes longer. As you've seen me say before on the topic of trophies, there is no guarantee with random chance, and for everyone who gets their last gem in a day or two, there is someone who is stuck for months. Waiting isn't easy, but while you do, you can be earning KinzCash by selling duplicates to Arte, or doubling up on gems to get your next Crown of Wonder that much sooner!


Final Thoughts
I realize that your cognitive dissonance bias is busy trashing everything I've just said. But I'm a fan of reason and a hopeless optimist. In the end, although there are a lot of mistaken tips passed around, they don't really hurt you as long as you look in the right mine. If the well-meaning but unfounded hints are truly irrelevant, they are so whether you follow them or not. Wheeling that cart back and forth in the mine is part of its charm, so by all means take your time and explore.

The cost of superstitious tips isn't always zero in WW, or the RW (cognitive dissonance is heavily studied in the context of financial markets for example), so there's an interesting lesson in the way the mines make you see patterns that aren't there. We see the world as we are, not as it is. Our experiences, prejudices, assumptions, and even the wiring of our brains and sensory organs together create our own little personal worlds. This is why we should look twice before stepping into traffic, and it is why we should think twice before dismissing someone else's perspective. Reality is relative. If something as simple as Arte's Gem Hunt can trick us so easily, what else do we think we know?

Webkinz World can be a wonderful place to learn, even beyond the features that Ganz intends to be educational. All you need to do is pay attention to what you're seeing, understand it, and talk about the difference between what is real, and what is not.

Last edited by williamson_39; 12-19-2008 at 03:36 PM..
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Old 01-28-2008, 01:16 AM   #2
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Default Re: Arte's Gem Hunt - EXPLAINED !!

Your brilliance shows once again. You and game theories, fits like apples and cheese. That is really good if you've never tried it . You are truly amazing, and thanks again for a guide on every game! They must have taken you forever.

You rock,
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Old 01-28-2008, 02:05 AM   #3
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Whoa! I am so impressed! Reminds me of how my 16yo asperger's syndrome son thinks! And he's almost always right, too! Rep points & thanks again!!
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Old 01-28-2008, 08:59 AM   #4
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Default Re: Arte's Gem Hunt - EXPLAINED !!

williamson_39, you rock! Or perhaps I should say that your posts are those rare sparkling rocks that have a gem inside each time without fail? LOL. Last Friday, I noted your posts on ennazusjsc’s wonderful thread with great interest (with the useful spreadsheet tool); happy to see you were indeed able to double check the probabilities and have had time to finish this first-rate myth-busting write up.

We have a natural tendency to filter our perceptions to discard inputs that contradict our existing ideas, and in the process ignore failures and over-emphasize successes. We don't know we're doing this, but we love being right.
Well said. I’m either not doing the Gem Hunt every day or my cognitive dissonance bias is in play: it seems that I hardly ever leave Arte’s with nothing but slag, maybe only two or three times a month (which is much lower than the theoretical expected value of 7.5 slag-days per 30 plays). Interesting. Either I am lucky -or- I am ignoring my failures -or- am doing the Gem Hunt only half the time.

It means that 50% of the time, you will find your last missing gem in two weeks or less. Nine times out of ten, you'll be stuck with one gem remaining to finish your collection for less than a month and a half.
You are indeed an optimist to have stated this so encouragingly! Sounds much better than
"1 in 10 of us will be looking for more than a month and half, and a whopping 1 in 27 will be looking for more than two months!"
For that 1 in 27 that will be looking for two months (or even that 1 in 5 that will be looking for more than a month, or 1 in 10 that will look for 1 1/2 months), perhaps you could add a little chart showing how much money we could expect to make if we sell all the gems we find while looking for that last one? That will help us feel better about it taking so long to complete our Crown of Wonder.

p.s. Math looks fine…although to be truly correct you ought to use the approximately-equal symbol (which I couldn’t get to work in WI) or use fractions. But it’s probably good to round to two-decimal places: a wise engineer will not scare away people with too much math…or else people that have been unsuccessful with math will flee in great numbers! [I believe it’s the cognitive dissonance bias at work again. Too many people convince themselves that they are bad at math at an early age, then spend their life ignoring evidence to the contrary.]
p.p.s. I did send a PM about the intermediate calculation. More later.

Last edited by jillion; 01-28-2008 at 12:50 PM.. Reason: added p.p.s.
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Old 01-28-2008, 10:35 AM   #5
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Default Re: Arte's Gem Hunt - EXPLAINED !!

Okay -- so I will get that lat gem withing a month and 1/2 or you are going to have to trade me for it.... Just kidding -- a lot to absorb for me -- read 2x
before got most -- lol (your math geneious is showing) congrats on another great thread....
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Old 01-28-2008, 10:51 AM   #6
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Default Re: Arte's Gem Hunt - EXPLAINED !!

View Post Originally Posted by jillion
…although to be truly correct you ought to use the approximately-equal symbol (which I couldn’t get to work in WI) or use fractions.
I agree. I didn't really pay enough attention. But I did use full detail to do the calculation and I tested it with the rounded numbers I showed anyway, just to make sure that the rounding error was not going to contradict me. I think I'll modify it to use fractions, and draw a picture to show it instead of letting all the fractions tip over like "7/18" (I can't use the approximately equal symbol without making a graphic anyway). Now that my formula has passed muster, I can invest the time instead of wasting it....

You've got a good point about glossing over the realistic possibility of a longer wait. I think I can do both - optimistic and realistic don't have to be mutually exclusive, maybe.

Thanks for your help Jillion - you're amazing. Even your suggestions for ridiculously complicated thought-provoking additions. The potential-earnings angle I like, but I'll have to think about it....

---------- williamson_39 added 10 Minutes and 43 Seconds later ----------

By the way, you'll start to notice those slag days when you're down to one or two gems left.

Last edited by williamson_39; 01-28-2008 at 11:30 AM.. Reason: Automerged Doublepost
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Old 01-28-2008, 11:38 AM   #7
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Default Re: Arte's Gem Hunt - EXPLAINED !!

BRAVO!!! BRAVO!!!

and thank you SOOO much for releasing me from my preconceived notion that it mattered which rock I picked!!!
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Old 01-28-2008, 01:35 PM   #8
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Default Re: Arte's Gem Hunt - EXPLAINED !!

Thanks to jillion for catching an error in the math. I think the logic was right, and the result is close to the same, but I've edited the main post to cover my tracks.

LOL, I was caught with my head on backwards, but hindsight saved me.

Last edited by williamson_39; 01-28-2008 at 01:41 PM..
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Old 01-28-2008, 09:21 PM   #9
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Default Re: Arte's Gem Hunt - EXPLAINED !!

Well, well. You finally did it! A very thorough explanation. I'm also glad to see that we've got the same consensus on my spreadsheet as we did before and don't have to change it again after reading this thread. On the other hand, could I change equation 2 to address the probability of finding one of the many gems I need in a particular mine by replacing the P(one gem) with P(one of the four needed out of fourteen gems)? Would the math be the same with the substitution of 4/14 instead of 1/14 for example? I could then actually compute a success probability for each mine in my sheet. I don't think it would change the resulting suggested search location, though it would be a cool addition.

By the way, thank you for all your helpful comments on my thread. I'm so glad I could help so many with it and your suggestions have only made it better. YOU ROCK!
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Old 01-28-2008, 09:50 PM   #10
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Default Re: Arte's Gem Hunt - EXPLAINED !!

Man, you SO trashed my theory...yes it was a cognitive dissonance bias (ok, ok) I REALLY did believe that if it 'sparkled' the right color I was getting the right gem. However, in the last few tries it has yielded nothing but slag. This does calculate it out better (let's face that fact) and I think it shows the odds as greater than I had thought for a gem of any nature. I am going to be putting it to the 'test' (though I am sure it will only serve to prove correct,) tomorrow. I am anxious to try this. I will say that I am annoyed with the , though. I have hunkered down in that Buried Bones mine for the last two weeks trying to get it and it has been quite elusive.
I might add at this point, too. You spoke about selling off your duplicates. While you are at it, don't forget to tip the vendor. One thing I DID notice is, that the higher you go on his friend scale, the better the deal he gives you, even on the low end stones.
In the end, it is the thrill of the hunt, the grand conquest! I WILL get that one of these days. It is my great Moby!
Thank you for some excellent insight that is applicable in both VW and RW. It is amusing how we 'play the odds' in both.
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