Random Trophies in Webkinz World

williamson_39 · 01-15-2008, 04:54 PM

Patience, or Patients?
Williamson_39

When it comes to the random trophies in WW, there are two big questions:

What do I have to do to get one?
How many times do I have to play before I win?

The first question was answered in the Random Trophies...EXPLAINED thread (sticky in this forum).

The second question is tougher. If you play "a lot" and don't see a trophy, you start to wonder if you're missing something. The odds in the above thread don't give you an intuitive feel for how many times you should expect to have to try (unless you're very familiar with probability). I've seen many many questions about one game or another, and everyone feels the same way with random outcomes. It's just the way we're wired (or weird, as the case may be). A ten game drought has you wondering if Ganz changed the game, or suspecting that your account is jinxed. I know the feeling.

So I've done a little calculating, and I think I've come up with something that might help. Don't be put off by its appearance of complexity - I don't want to hear how terrible you are in math (shush now, I mean it!) - that's not a good way to start. Bear with me and give it a chance. If all goes well, in another five minutes you'll understand these games better than Ganz does.

Likelihood by Games Played
Each game has its trophy-chance task. Depending on how many times in a game you can do that task, the number of games you'll play for a trophy will differ, over the long run. But in order to keep this relatively short, I've made some assumptions about an average game in each case. Operation Gumball is a bit different, so I've made two different calculations, one if you routinely get past level 7, and one if you usually finish level 19 (more than that, and you certainly don't need my help).

Here's a graph of the results:

The vertical axis shows the number of games played, the horizontal axis is the likelihood of winning a trophy. The Games Played axis is a special type called "logarithmic" - you can see that the minor tick marks are not equally spaced - this allows the graph to cover a larger range and still be readable at both extremes (on a 'normal' graph all you see is a big curve for Go Go Googles and everything else squashed down at the bottom). Each minor tick mark counts up by the amount shown in the major group below it (1,2,3,...,10,20,30,...,100,200,300,...). I hope some examples will help you read the graph.

Cash Cow (1 in 100 chance per group clicked over the minimum required for that level)
You have a 1 in 10 chance of winning a Cash Cow trophy in only 2 games! (The red line on the graph is for Cash Cow; it crosses the 10% level at the 2 games played line). By the time you've played 12 games, you have a 50-50 chance of having won the trophy. Nine times out of ten (90%) you will win a trophy in 39 games or less. All of this is true only if you can manage to click six groups that meet the minimum size per game.

Candy Bash II (1 in 100 per cat, if unlocked with a 4-of-a-kind combo)
If you can unlock the trophy chances four times a game (and if you can catch), half the time you'll see the trophy in 18 games or less. Nine times out of ten you'll see it by your 58th game (the yellow line for Candy Bash II crosses the 90% mark just below the 60 games line). If you can unlock more chances per game, the number of games required could be much fewer.

Booger Gets an A (1 in 50 per filled report card)
Since most players can only complete one card per game at best, this one is easier to estimate. You'll need to fill the card 35 times to have a 50-50 chance at the trophy (Booger's line crosses 50% likelihood between 30 and 40 games on the graph).

Goober's Lab (1 in 400 per 5-atom bonus popped)
Big brother Goober is much less generous, but luckily you can get more chances per game. If you can give yourself five chances per game, half the time you'll see the trophy in 56 games or less (the line for Goober's crosses the 50% level about half-way between 50 and 60 games played).

Operation Gumball (odds improve the higher you get)
As I said, this one is a bit different. You can see that it can be among the most common (if you play well), or the most rare. The most important thing to note is that this is one game over which YOU can have a big influence on your chances. Read the tutorials and practice. It will be a 'cash cow' or a googles long shot, depending on how well you play.

Go Go Googles (1 in 400 each time you spell 'Googles')
What can I say? Don't be surprised if it takes you a long time. Even after you're surprised, you'll still be waiting.

**NEW** Likelihood by Playing Time
Here's another view, based on a great suggestion by jillion in a post below. It didn't actually take too long, and I came out of it with three trophies (Cash Cow, Booger, and, wait for it, Go Go Googles!).

In this graph, the vertical axis measures the playing time instead of just the number of games. The times are based on my own play, so will vary from yours, but as jillion pointed out, the relative position of the trophies may be telling. I first calculated a value for how much time it took me to earn a trophy chance in each of the games (averaged over several games and multiple chances). I'm sure I did NOT do enough trials to be sure these were reasonably stable, but it worked out like this:

Go Go Googles: 5 minutes (slow strategy - see the Googles guide for a new alternative)
Operation Gumball (level 7): 4 minutes
Operation Gumball (level 19): 14 minutes
Goober's Lab: 1.5 minutes
Booger Gets an A: 6 minutes
Candy Bash II: 0.8 minutes
Cash Cow: 0.5 minutes

The resulting graph (using the same colour scheme) looks like this:

The Go Go Googles line is still depressingly high of course. If you played a 24-hour marathon, spelling "Googles" every five minutes the whole time, you'd still only have a 50-50 shot of seeing the coveted prize once. It is possible to play it faster, but there's no free Goo Goo Berry lunch.

There are a few other interesting (and happier) things about this graph. Operation Gumball is quite remarkable in that the lines for the two alternatives (finishing level 7 or finishing level 19) converge significantly compared to the Games Played view where they were at opposite ends of the spectrum. This means that even though finishing more levels improves your odds, the time required to play those extra levels makes a big dent in that advantage in terms of the cost of earning the chance. I think I play quite quickly too, so it could be even less advantageous.

The message there is that you don't have to be a master player to win in a similar amount of time. That's actually the whole point of offering random trophies - they're accessible to a much broader range of players. They're also frustrating, but I think knowledge always engenders greater patience.

I have to mention too that this points to a tradeoff between score (KinzCash and hypothetical tournament placing) and "trophy investment efficiency". If you quit a game early, you are often giving up the sweetest KinzCash returns late in the game. But if you continue a game beyond where you will earn another trophy chance, you are wasting time from a trophy perspective. Losing Go Go Googles with one letter missing is BAD. Losing Booger gets an A with five checkmarks is painful. So if you really want to get a trophy, quit after your last reliable trophy chance, and start over. Of course, not everyone wants to be so mercenary about it. I'm not telling you to spoil your fun!

Final Thoughts
Note that there is never a guarantee! I don't show "100%" on the graphs because the lines turn north and go forever. The games don't know that you've tried once, or 500 times, and each trophy chance is independent of all before it. All you have in your favour is increasing likelihood the longer you go without winning. Most likely, you will never play a few hundred games without winning a Cash Cow trophy, just as most likely you will never scratch a winner in a Quick Draw event.

An interesting side-effect of the graphs (particularly the new Playing Time version) is that it gives you an idea of what the relative trading value of the trophies should be. The exception may be Candy Bash II for a while, until the new info about that game makes the trophy more common for everyone. As I said however, my performance may not be representative, and games (and therefore trophy supply) are also driven by popularity, so effort can't be a perfect gauge.

If you go to the WW newspaper and click the "Scores and Achievements" link, you'll see a tab on that page called "My Scores". It shows the number of times you've played each game, as well as your average and high score. It might be interesting to put a tick on the graph each time you win a trophy, depending on the number of games since your last win, or playing time if you keep track. The ticks should eventually balance around the 50% likelihood level (although for some games which shall remain na na nameless, you won't win enough trophies to see any trend).

Be sure to check the original Random Trophies...Explained thread if you haven't lately, because it now contains links to guides for most of the games that will help maximize your opportunities. You may need to adjust your position on the graphs depending on how many chances you usually earn, but based on that, you can follow your game's coloured line and decide if you really are unlucky, or just impatient!

Either way, I wish you all the best.

commodore · 01-15-2008, 04:59 PM

Like all your threads, this one is just as impressive. Thanks for working so hard, you are an asset to WI.

littlex3red · 01-15-2008, 07:59 PM

Very impressive -- got it -- Totally appreciate the information -- Keep up the great work and thank you so much -- I think I am the patient.

jillion · 01-15-2008, 10:58 PM

Most excellent, Williamson_39! You’ve outdone yourself. I like that you have made your graph so visually pleasing and have so nicely outlined how to read the graph in such a clear, easily accessible manner. The walk-through is particularly nice. I think the odds are good (better than 9 out of 10) that a randomly-chosen WI user will be able to interpret the graph if they read through your explanation.

Originally Posted by williamson_39

... An interesting side-effect of this view is that it gives you an idea of what the relative trading value of the trophies should be. The exception may be Candy Bash II for a while, until the new info about that game makes the trophy more common for everyone.

The correspondence with the trading value of trophies is the second thing I thought of when I saw your graph (after noticing how beautiful it was, of course). Actually, the current trading values in the WIGEV guide seem to match up nicely with the graph, with the exception of Booger’s and Goober’s (Booger’s is probably too high only because it is newly brought back from the land of retireds; Goober’s is a bit undervalued perhaps because it’s easy to become addicted to it, burst more than 5 atoms, etc.). Operation Gumball is as you say, in a league by itself…it’s sort of a hybrid between the random trophies and the non-random trophies, since the odds change so much with skill of the player.

So, now I would like to see two things:
1) a graph of where everybody’s at in terms of their trophy collections. Say, maybe 1000 WI users data for each game (?). [Okay, probably unreasonable.]
2) a graph that takes into account the time needed to play the game. Specifically, a graph just like this with the vertical axis measured in hours of play instead of number of games. Ideally this would correspond very closely to the trading values of the trophies. For instance in Goober’s: bursting the a spinning atom only 5 times can be done at a fairly low level (I’d say by Level 3 or 4?), so that would make those 5 trophy-chances happen pretty quick (compared to 19 Levels of Gumball).

In the meantime, keep up your wonderful habit of bringing logic and reason to WI! Your fantastic rainbow graph will certainly help us all feel better about trophies and may even help convince more people that math really is cool.

Dipman · 01-16-2008, 02:33 AM

Again, another amazing post. Thanks for all of that information. It helped reassure me that there is no massive conspiracy against me by Ganz!

williamson_39 · 01-16-2008, 02:49 AM

Originally Posted by jillion

Your fantastic rainbow graph will certainly help us all feel better about trophies and may even help convince more people that math really is cool.

Thanks! I spent a disproportionate amount of time on the formatting, so I'm very happy you noticed.

I had a hard time deciding whether to focus on games played or actual trophy chances. I really like your suggestion about playing time too, but that one has its own problems.

The reason I settled on games played was that I felt that people naturally overestimate their trophy-earning opportunities, and they can more readily track games played. I thought a deliberately conservative estimate would make the graph easier to relate to (and feel superior to) - and could in most cases be taken as "x number of games, at most." Some of the estimates were difficult - Goober's in particular is quite low, I agree.

The playing time aspect would definitely provide a better measure of "effort", and thereby should be a better value guide, all else being equal. But then again, chances per unit time would vary by player too in most games - so all else isn't equal. Assuming that pixel-trading is rational enough to correlate is a bit of a stretch too, come to think of it.

For example, my new approach to Go Go Googles improves my spellings per hour by as much as 30%. But it throws off my games played (I often only spell once) and hopefully my playing time per trophy. Fortunately, I can do the math. I guess you could turn it around and use the odds and resulting trophies to calculate what the average chances per hour must have been. Operative word being "you".

It would be cool to have access to Ganz's own playing stats. I think that's the only way it could be dealt with exhaustively, although they may only track plays and trophies. There are so many factors influencing manual submissions that I'm afraid the variation in input would have as big an impact on the results as my arbitrary estimates. Ah well, we wouldn't want WW to be fully determinate anyway - a little harmless mystery leaves room for imagination.