On a whim, I decided to start tracking tribute rates. All my kingdoms are at level 10, with 5+ stars, my guild’s red statue is at level >100. Here’s a table with the totals as of this evening:
|Kingdoms||# Times||Expected #||Difference|
That right hand column is the difference between what I expect based on probability (26% for each kingdom, times 29 kingdoms each time I collect tribute) and what I actually got. Ignore the loony number of 7-kingdom tributes; that’s at least close to the 7.5 average I expect. Instead, pay attention to how skewed the distribution is. 4-7 kingdoms (i.e. below average) are all coming up more frequently than they ought, and 8-11 (i.e. above average) are all coming up less frequently.
In fact, the average success rate, treating each kingdom as an independent Bernoulli trial, is 24.8% over 4611 trials. This is lower than the 26% I expect. It is, somehow, actually an improvement over the first 28 tribute collections (812 trials), which paid out at a dismal 23.2% rate.
Now, pay attention as I (probably mis)apply statistics. If the kingdom tributes aren’t being awarded with a 26% chance, what probability is consistent with my results? I use the Agresti-Coull interval, which I discovered on Wikipedia, to estimate the true probability associated with my results.
4611 trials, 1144 successes
alpha = 0.05 (corresponding to 95% confidence interval)
p_low = 23.6%
p_hi = 26.1%
So, based on these results, I am just barely within the 95% confidence interval that the tribute distributions are actually being awarded with a 26% chance.
More data is better. What have you all observed recently, or in the future? For ease of compiling data, I’d prefer to hear from late-game players like me who have a uniform 26% across the board. If someone who knows more stats wants to deal with combining Bernoulli trials at different probabilities of success, more power to you.