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 |
---|---|---|---|

0 | 0 | 0.03 | 0 |

1 | 0 | 0.26 | 0 |

2 | 1 | 1.29 | 0 |

3 | 5 | 4.07 | 1 |

4 | 11 | 9.28 | 2 |

5 | 19 | 16.3 | 3 |

6 | 25 | 22.9 | 2 |

7 | 37 | 26.5 | 11 |

8 | 21 | 25.6 | -5 |

9 | 16 | 21 | -5 |

10 | 11 | 14.7 | -4 |

11 | 3 | 8.94 | -6 |

12 | 7 | 4.71 | 2 |

13 | 2 | 2.16 | 0 |

14 | 1 | 0.87 | 0 |

15 | 0 | 0.31 | 0 |

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.