Evolution of the Phoenix

[OldHag]

It's pretty clear you just threw these extra quests in without much thought. Probably recognised there were gangs of us completed the quests and sat around twiddling our thumbs waiting for wishing well day, I don't see a big rush on for the daily prizes....rather lame (imho)
But being that you decided to add them, don't you think you should increase the sky essence for them.....quest 114 is 64 essence so far ALL of these new ones are 65 whilst being clearly more difficult (time wise) to complete......Come on, it's not to late to increase the amount of essence, particularly if you added these so that we have a better shot at winning more prizes......wink, wink!

 

[DeletedUser5093]

After this latest update I thought I'd run the numbers to see how many feathers are possible based on what strategy is being used. Thought you guys might be interested, so I've summarised it all below. Let me know if you see an error, have a question, or want me to explain a step in a bit more detail.

Spreadsheet
To see the details of all the below calculations you can view or download a copy of my spreadsheet here.

Total SE
With these new quests available we are now able to get a total of around 6671 to 6991 SE over the course of the event, assuming you complete all quests, and taking into account the variable amount of SE you can pick up around your city. I've lifted these numbers straight off ElvenStats.

To figure out how many feathers we'll get we take the total SE we'll get, and divide it by the average feather cost of all the chests we select. To calculate this we use the effective 'Sky Essence Per Feather' figures found in the 4th column of the table at the very bottom of this page and then adjust it based on the probability we'll actually pick that chest. The latter will depend on what strategy we use.

Strategy 1: choosing a random chest every time.
If we pick our chests at random each time then we can assume every chest will be picked roughly the same amount of times each, so the typical amount of SE spent on each feather works out to be the average of the effective feather costs of the 9 available chests, which is 23.24 SE per feather. This is the same figure ElvenStats uses in their own calculations to figure out how many phoenix artefacts will be obtained. This results in 287 to 300 feathers, or 7 to 8 artefacts. This method will also net you about 19 or 20 daily prizes.

Strategy 2: always choosing the most cost effective chest for feathers.
This method will maximise the amount of feathers you get. Essentially it involves always picking the chest with the cheapest feathers (see above link to table on ElvenStats). Since some chests will be picked much more often than others this calculation is non-trivial. It involves figuring out every possible combination of 3 chests you'll get, then figuring our which chest you'll pick with this strategy, then taking an average of the feather cost of the chosen chests for all those different scenarios. This works out to be 20.26 SE per feather, which will get you 329 to 345 feathers total, or 8 to 9 artefacts. Note that 9 is enough to upgrade a single phoenix to the maximum level. You can also expect to gain about 19 or 20 daily prizes with this method.

Side note on Strategy 2:
Previous calculations by myself and others have used a much simpler calculation, which involved simply taking an average of the 7 best chests (assuming the 2 worst ones will never be picked). This predicted 300 to 315 feathers, or 8 artefacts. I now believe this calculation to be incorrect, as it doesn't take into account the frequency with which some chests will be picked. Just something to be aware of in case you find these numbers floating around still.

Strategy 3: always choosing the most cost effective chest for daily prize.
This method will maximise the amount of daily prizes you get. Since not everyone is focused on the grand prizes I thought I'd run this scenario as well. Essentially this method involves picking chests based on the cheapest 'SE Per Daily Prize' value (column 5 in the table linked to above). This calculation is similar to Scenario 2, except a different chest is chosen from all the various possible scenarios. The average feather cost of these chests works out to be 23.41 SE per feather, which will get you 285 to 298 feathers, or 7 artefacts (maybe 8 if you're lucky). You can also expect around 25 to 27 daily prizes.

Summary
Assuming you have average luck, and you complete all quests, you should pick up at least 7 artefacts regardless of which method you're using, which is enough to get a level 8 phoenix. If you pick a strategy to maximise the amount of feathers you get then you can potentially get enough artefacts for a level 10 phoenix without needing to use diamonds. You can also expect around 19/20 daily prizes if you either pick chests at random or optimise them based on feathers, or as much as 25 to 27 if you're instead focusing on daily prizes.

I hope you guys find this useful.

 

[FieryArien]

However, I do want Phoenix evolved as much as possible, so my demolition team simply had to find space elsewhere.


I have to ask, do we? Who benefits from these prizes, and this event?
My smaller cities are not yet topped up with relics. I would have a VERY hard time with my tier 4 and 5 boosts at this point BECAUSE the only way to get an Orc Nest is craft them (IF it's offered which half the time it isn't); for example. All this drama for the opportunity to run our cities into walls time and again.
These quests are ridiculously hard, nearly impossible for small cities, all for a temporary effect. Consider: relics for pet food for bonuses that don't last. Tsk tsk

Nonono, don’t get me wrong. We certainly don’t have to. I was merely expressing my preference: I do want Phoenix (...), so my demolition team (...)

I didn’t get the middle paragraph about small cities, tier 4&5 boost and Orcs Nest. :-( Anyway, in my case my smaller city is in Orcs, so I don’t consider it small anymore. And it has no problems with relics at that stage. Anyway, for that city the newly added quests are bearable, because I was lucky to get marble boost (small factory).

The thing is I’d like to have an expansion or two free at all times for playing around during events, FA, challenges or simply for rearranging. But even though I have several premium expansions, I’m failing miserably in achieving that!

 

[DeletedUser3097]

Please ensure that all further contributions to this thread are about the event Evolution of the Phoenix. If you wish to discuss another player's playing style with them, use the personal messaging system. Other Forum Users may have different viewpoints from your own however it is essential that all replies are respectful of this. The Forum Rules outline what is and what is not acceptable.

 

[DeletedUser7130]

I have a question: Has anyone received Essence rewards from chests? I have opened about 30 chests that were supposed to give 10 or 20% chance of getting Essence. Not a single one so far. Am I just really unlucky, or is there something wrong with their numbers?

 

[Deleted User - 1759805]

Strategy 2: always choosing the most cost effective chest for feathers.
...
Strategy 3: always choosing the most cost effective chest for daily prize.
...

Great job there, thanks! Now, I expanded the analysis a bit, and looked at how much difference S2 vs S3 will make in terms of daily prizes. I should have done it in terms of probabilities, but I just averaged scores - it should be quite indicative.

So If you focus on best chests for dailies (i.e. S3), your average score is ~258 essence/daily prize. If you focus on feathers, it drops to ~348. So basically, you need to spend ~33% more essence to get the same number of daily prizes. If we assume about 7,000 total essence max, you're choosing between ~27 vs ~20 daily prizes, on average. So you're making a choice between ~1 extra artifact vs ~7 daily prizes.

Now, before you get overly excited, I was following the strategy for daily prizes on day 3, and my realized score over ~1,700 essence was 423 - much worse than expected 258. These averages have a lot of variability around them, and you can always get the short stick.

 

[Deleted User - 341074]

all for a temporary effect.

I haven't checked at lower chapters, but at higher ones, an evolved phoenix gives a HUGE amount of supplies and decent pop/sq. Those are permanent effects.

 

[Pauly7]

I have a question: Has anyone received Essence rewards from chests? I have opened about 30 chests that were supposed to give 10 or 20% chance of getting Essence. Not a single one so far. Am I just really unlucky, or is there something wrong with their numbers?

Quite a few actually. It probably is just luck. The other day I played 2500 sky essences and won the daily prize 21 times, which meant 84 x 14 hr time boosters. Lots of luck along the way including hitting the 200 bonus SE quite a few times.

 

[Deleted User - 1759805]

Quite a few actually. It probably is just luck. The other day I played 2500 sky essences and won the daily prize 21 times, which meant 84 x 14 hr time boosters. Lots of luck along the way including hitting the 200 bonus SE quite a few times.

And I've got only 4 for 1700 essence...

But are you sure about your numbers? It indicates that you've spent about 120 essence per successful daily. This is almost twice as good as even the best chest average (~206), and given large number of draws you being so far off average is quite a low-probability event. Do you know how many chests you have opened? Curious about your overall success percentage was. Mine was 10% (4/40), which is pretty bad considering all the chests that I opened had at least 10% chance of success - and many were better than that.

 

[Deleted User - 1759805]

Quite a few actually. It probably is just luck. The other day I played 2500 sky essences and won the daily prize 21 times, which meant 84 x 14 hr time boosters. Lots of luck along the way including hitting the 200 bonus SE quite a few times.

Yeah, something is really fishy here - or you're just extremely lucky. I mean extremely!

Spoiler: DANGER: some math ahead!

Probability of achieving M successes in N trials, where each trial has probability of success of P, and trials are independent of each other follows binomial distribution. So here we know that M=21 (21 successes). We don't know aggregate P and N, but let's assume the best case scenario, and assume that you only opened the best chest for dailies (purple) on every draw. This is unrealistic, so the actual result would be worse, but still. Then P=30%, and number of trials N is ~30 (2500/82). Now, you can type these numbers into any binomial probability calc, and it will spew that probability of hitting 21 dailies or more in this setup is 0.000727783%, or worse than 1:100,000. Remember, these were very optimistic assumptions.

TL;DR: Probability of hitting 21 dailies or more in this setup is less than 0.0007%, or worse than 1:100,000. You should try lottery

 

[DeletedUser7130]

Quite a few actually. It probably is just luck. The other day I played 2500 sky essences and won the daily prize 21 times, which meant 84 x 14 hr time boosters. Lots of luck along the way including hitting the 200 bonus SE quite a few times.

Thanks for the reply. I'll just keep hoping then

 

[DeletedUser5093]

Quite a few actually. It probably is just luck. The other day I played 2500 sky essences and won the daily prize 21 times, which meant 84 x 14 hr time boosters. Lots of luck along the way including hitting the 200 bonus SE quite a few times.

Probability of hitting 21 dailies or more in this setup is less than 0.0007%, or worse than 1:100,000. You should try lottery

Assuming Pauly's strategy is to pick the best chest for daily prize every time, then my calculations give a scenario of 21 daily prize wins as exactly 3 standard deviations above average, i.e. better than 99.86% of players. This equates to about a 1 in 740 chance of occurring. So extremely lucky certainly, but not outside the realm of possibility. (Using the lottery as an analogy, this is about the same chance as picking 4 of the 6 winning numbers, which might net you about $25 in prize money if my local lotteries can be considered typical. Don't quit your job just yet. )
Not sure where the 1 in 100,000 figure is coming from.

 

[Pauly7]

Yep, certainly sure about the 21 daily prizes. I didn't count exactly on SE used, but it was certainly no more than 2,700.

@UlyssesBlue thanks for the numbers. The strategy was officially to go for the feathers, though I found myself leaning more on the 82 chest than the 18 or 20, when it was available, purely for my run of luck hitting 200 free SE. (Interestingly, I only hit the 7 x MM spells 3 times, by comparison).

One in 740 sounds reasonable. Before seeing that I was going to say off the top of my head that perhaps it's 1 in a 1000, but even if it is then there is someone out there playing that will get that return.

On the flipside of this I have had several occasions in the past of yelling at my computer screen and saying that "this is just not possible" after having a run of bad luck. As I have said before when that happens to people then they get on the forum to shout about it, but not so often with comparative runs of good luck.

I had one occasion on last year's Autumn event where there was one particular daily prize that I knew was going in my city and I had made a space for it and everything - I ended up rolling just shy of 2,000 before I eventually got it, cleaning out everything I had and scuppering all my plans of subsequent daily prizes.

 

[Deleted User - 1759805]

Assuming Pauly's strategy is to pick the best chest for daily prize every time, then my calculations give a scenario of 21 daily prize wins as exactly 3 standard deviations above average, i.e. better than 99.86% of players. This equates to about a 1 in 740 chance of occurring. So extremely lucky certainly, but not outside the realm of possibility. (Using the lottery as an analogy, this is about the same chance as picking 4 of the 6 winning numbers, which might net you about $25 in prize money if my local lotteries can be considered typical. Don't quit your job just yet. )
Not sure where the 1 in 100,000 figure is coming from.

I have posted the math in the spoiler above. How did you come up with 3 sigma?

If I re-interpret that Pauly's 2,500 SE is not the amount of all opened chests (as I have read it), but initial SE balance which is used until it's zero, then calculations are different. The difference is SE gained on opening the chests - in the second case the total value of opened chests can be significantly higher than 2,500. So under second assumption we can use adjusted cost for chests (i.e. 62 instead of 82) - meaning in the all purple example we can do 40 chests instead of 30 (basically, you expect to get 4x +200 SE draws).

In this case, 21 successes out of 40 draws is about 1:400 chance. If you get lucky with +200 draws, you can do even better than that. That's certainly possible, but that's for the all purple scenario which is unrealistic. The next best chest - green - is significantly worse.

Bottom line is, probability of runaway success is highly dependent on getting extra SE from chest openings. If my re-interpretation of Pauly's setup is correct, he is still very lucky, but indeed probably should keep his job for now

 

[Timneh]

In the carnival event i had fantastic luck and got all the daily prizes i wanted first or second time on every one. Sadly on this event i think @Pauly7 must have got all my luck lol. I am getting them but i am having to open a lot of chests to do it, now building up SE ready for the next one i want. I just think oh well, i can't be lucky every time.

 

[DeletedUser5093]

How did you come up with 3 sigma?

It sounds like we used very different assumptions and methods in our respective calculations. I assumed the '2500' figure referred to the initial SE before the first chest was purchased, and also accounted for SE won from opening chests. In the spreadsheet I posted in my earlier analysis I determined every possible combination of 3 chests, and which chest would be chosen in each scenario if aiming for daily prizes, then listed the corresponding SE per daily prize for that chest (values sourced from ElvenGems). I then calculated both the average (257.8) and standard deviation (46.2) of this list of values. Given an initial stockpile of 2500 SE, I then calculated how many daily prizes would be won for several scenarios: average, average -1SD, average -2SD, etc. Average - 3SD corresponds to 20.98 daily prizes.

 

[Deleted User - 1759805]

It sounds like we used very different assumptions and methods in our respective calculations.

It seems that neither of us had a good methodology So I just slapped together a full simulation - random chest draws, open chest selection, random open chest results etc (including extra essence). There are no additional questionable assumptions here, except that underlying random draws are independent. So here are the results (over 10mm simulations, so results are pretty stable):

Choosing best chest for daily prizes:

Average chance of daily prize per chest:                17.2%
Average amount of essence per daily prize:              241
Average value of chests opened:                         49
Average number of chests opened for 2,500 essence:      60
Average number of daily prizes for 2,500 essence:       10.4


Chance of hitting 21 daily prizes or more:       0.0732%
Chance of hitting 21 daily prizes or more:       1:1366

Choosing best chest for feathers:

Average chance of daily prize per chest:                11.0%
Average amount of essence per daily prize:              285
Average value of chests opened:                         34
Average number of chests opened for 2,500 essence:      79
Average number of daily prizes for 2,500 essence:       8.8


Chance of hitting 21 daily prizes or more:       0.0061%
Chance of hitting 21 daily prizes or more:       1:16420

Top 5 numbers for each strategy can be derived from your spreadsheet analytically, without simulation - and match pretty well, so I am reasonably certain the simulation works correctly.

Interesting observation is that while expected numbers of daily prizes do not differ that much (10.4 vs 8.8 for Pauly7's case), probabilities of knocking it out of the park (such as 21 dailies or more) differ by more than a factor of 10x.

@Pauly7: It sounds like you employed a hybrid strategy (focus on feathers but over-weighing purple chests), so for you probability of hitting 21 (or more) would be somewhere between numbers for these 2 pure strategies. Still, I'd say you were really, really lucky - less than 1:1000

 

[Pauly7]

Interesting... but, does the 241 SE per daily prize factor in the chance of bonus 200 SE prizes?

Whichever way you dress it up I was lucky. Or alternatively the conspiracy theorists can consider whether the probability of daily prizes remains the same with time booster days!

I will probably end up rolling a similar number of SE on 30KP day. We'll see what happens. I'll probably get my payback by hitting it once or something. The goal will still be feathers though.

 

[Deleted User - 1759805]

Interesting... but, does the 241 SE per daily prize factor in the chance of bonus 200 SE prizes?

Yes. The way to interpret this number is that if you start with 2410 SE and use it all, you expect to get 10 daily prizes. In reality you will use more than 2410 SE as along the way you will get some extra SE from chests (it's not just 200 from purple, but it's the primary contributor).

 

[Sir Derf]

So, I've collected through quest 126 and have 5,802 Sky Essences, saving them up for Phoenix Cults as a byproduct of attempting to maximize feather production.

Initial Sky Essences spent, 5,800, 163 chest opened, 255 feathers gained, 22.75 SE/feather
Initial Sky Essences spent, 12 Grand Prizes (6 Phoenix Artifacts), 483 SE/GP
Roughly as expected.

I opened 25 Chest 16 times and got no 10% chance of 20 Sky Essences.
I opened 82 Chest 34 times and got 1 10% chance of 200 Sky Essences.
In total, 50 attempts and I received 1 10% chance occurrence. Really bad Random outcomes.

On the other hand, with 103 %5, 2 10%, 24 15% and 34 30% chances at the Daily Prize, I received 20 Phoenix Cults, a 12.27% return from an expected 11.75% average, so roughly as expected.


I still have maybe another 1,350 or so Sky Essences to collect, so should get a few more Grand Prizes, hopefully to include 2 more Phoenix Artifacts.

Woo hoo.