Interesting thread.
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would we not expect the true value to sit closer to the middle of the 95% CI?
Would a Bayesian estimate of the distribution of p that best fits everyoneās data be symmetrical as well?
Given a sufficiently large data set to allow for the randomness to even out, the p value should approach 21.2%. The main takeaway from the smallish sample is that 26% falls well outside the confidence interval, so we can assume guild bonus is NOT additive.
New weekend plan: Trust falls.
Everyone quits their guild and records base Tributes for comparison.
Then hope our guilds let us back in before reset. 
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A few months ago, I ran some probability tables, 24 picks at 20% which is the same as 24 level 10 3 star kingdoms. Y axis is percent in 1% intervals of getting the X axis tributes. 0 tributes is 0.472% chance, or roughly 1 in 200.
But now I canāt remember what site I went to to build the tables.
The difference between 21.2% and 26% is significant and should be pretty easy to confirm with a limited number of trials.
The difference between 20% and 21.2% is minute, youād never be able to detect it without dedicated effort.
Looking at my data over 75 collects, my per kingdom chance is only 19.5%. While this does not preclude the chance that I could have been unlucky over the course of days, it does seem unlikely. Statistically speaking, there is only about a 5% chance I could get this poor a run over this long a time if the odds were actually 21.2%.
However, looking at the same data, and assuming that it only checks 26 of the 27 kingdoms, then that increases to over 25% chance of being correct, and my low scores are just bad luck.
Is it possible that Silverglade (or some other kingdom) is errantly left off the list to check for tribute? @Sirrian @Nimhain @Saltypatra
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I agree that is seems incorrect. Would really appreciate it if Devs would check and confirm it works correctly.
NowayJoe2Go
To my mind, Rasper has already confirmed it. His 95% CI leads me to confidently reject 26% even with the data he has.
Iām more concerned about the true value being located on the upper-side of his CI with his earlier data because - which makes me think it might not be 21.2, but a little lower.
I feel like I was seeing higher tributes before the last kingdom came out, but I wasnāt keeping track so it remains just a feeling. These kinds of things are easily done, Iāve made mistakes like this myself (* instead of +) or off-by-one errors.
100 tributes tracked now (image updated).
Iād need over 20 straight tributes of 8 kingdoms to bring the average up to the expected levels. The chances of getting an average of 19.3%, when the actual chance is 21.6%, over the course of 2700 samples is roughly 1 in 200. Either Iām perpetually unlucky, or something is off.
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Interesting. Hereās my list of tributes from when I was tracking:
0 kingdoms: 2
1 kingdom: 7
2 kingdoms: 8
3 kingdoms: 17
4 kingdoms: 21
5 kingdoms: 27
6 kingdoms: 26
7 kingdoms: 24
8 kingdoms: 9
9 kingdoms: 4
10 kingdoms: 1
11 kingdoms: 1
12 kingdoms: 1
Total: 148 events, 148*27=3996 individual tribute chances
758 successes, for a success rate of 18.9%
I appear to be even unluckier than you, and over an even longer period.
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Adding your data to mine gives a p value of 19.1% with a 99% confidence interval of 17.86 to 20.34, meaning odds off the real value being 21.6 is somewhere in the 1 in 3000 range.
edit: dropped a decimal place.
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Rasper - how are you calculating the ~1 in 30000 ? What assumptions go into that?
A quirk in calculating confidence intervals. Going from 80%, to 90%, to 95%, to 98%, to 99% each time the size of the interval is increased by about .16. Since the interval would need to increase by .9 to get up to 21.6, that would be 5 more steps. So it would be 5 steps of halving the confidence, which is about 1/32 of 1% or 1 in 3200.
Oops, dropped a decimal place. 1 in 3000 not 30000
edit: could be argued it is twice that, as the 1 in 3000 is just ānot in rangeā which would count above and below. Either way, we are talking awfully small chances based on roughly 250 tributes worth of data.
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By the way, I am almost positive one of my single-kingdom tributes was Silverglade. So if the list of kingdoms that can give tributes was only 25 spaces long, itās older kingdoms getting bumped rather than newer ones.
So itās a z-statistic assuming a normal distribution about the estimated p with a variance estimated by the standard error of pā¦ I knew this at one point, sorry, case of use-it-or-lose-it for me, thanks.
I have a couple of guesses regarding this:
- p is not being computed correctly, or at least not the straightforward way weāve been led to believe based on the in-game text
- the kingdom draws are not iid
- some kingdoms are not being counted
Have the devās even commented on this or verified how the percentage is supposed to work?
Are we sure they didnāt ninja nerf how tributes work to reduce them?
My numbers of tracking do not add up, just like the others. Iām not against a lower chance of this, Iām against the dishonesty of the system if the numbers do not match up. Where else would they be dishonest to us at that point.
Iām in no way claiming they are doing this on purpose, but the silence from them about this has me wondering.
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Before you break out the pitchforks, I do want to say that Iāve never gotten the impression that anyone on the dev team has ever been intentionally dishonest. Theyāve shown great generosity in the past when there was a problem, so I donāt know why they would (intentionally) change something to be less generous than it used to be. In short: follow Hanlonās Razor here, if this is in fact something deeper than a shared run of bad luck.
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Agreed, no conspiracy here - itās incredibly easy to make a mistake, especially when the result isnāt dramatically different from the correct one