## Introduction

Here I go again, trying to make sense out of the probabilities that have been thrown at us, and understand our chances of getting the staff at which cost.

Again: I am not a statistician, just someone who likes Excel and used to like probabilities in his student days. I might be wrong, so challenge me if you think you know better.

## Assumptions / Notes

Based on what I read on the wiki, I assume that the PVP roulette has the following odds:

- 20% chance of getting a feather

- 20% of getting a caducite

- 20% chance of getting a nugget

I will assume in the calculations that you will either do your 5 daily fights (getting 5 nuggets and a roulette spin) or not play at all for the day (so, I do not calculate days where you get 3 nuggets and no spin).

## Overall results

bad news, based on my model, it is unlikely that you will get the staff for free, even if you are a dedicated daily player

from: | to: | 25 | 20 | 15 | 10 |

0 | 0 | 29.8% | 0.0% | 0.0% | 0.0% |

1 | 15 | 23.6% | 0.0% | 0.0% | 0.0% |

16 | 25 | 21.7% | 9.9% | 0.0% | 0.0% |

26 | 35 | 14.5% | 16.1% | 0.0% | 0.0% |

36 | 45 | 7.0% | 22.4% | 0.0% | 0.0% |

46 | 55 | 2.5% | 22.3% | 1.1% | 0.0% |

56 | 65 | 0.7% | 15.9% | 4.3% | 0.0% |

66 | 100 | 0.2% | 13.2% | 72.9% | 0.9% |

101 | 200 | 0.0% | 0.3% | 21.7% | 99.1% |

How to read?

- The columns represent the nb of PVP roulette spins = the nb of days where you played, got 5 nuggets and 1 roulette spin.
- the rows is the gold you will need to spend
- the cell indicates the probability of the event

Example: if you play every day (25 spins = 1st column), you have 29.8% chance of getting all items for free (0 gold sto spend = 1st line).

Example 2: if you only play 15 times (15 spins = 2nd column from the right), there is 72.9% probability that you will need to spend between 66 and 100 Gold = more than a LE weapon...

But what is the average cost in Gold, and why is the variance so big between 25 and 20 spins?

25 | 20 | 15 | 10 | |

feathers & Cad | 15.7 | 26.6 | 41.7 | 60.1 |

Nuggets | 0.0 | 22.6 | 48.2 | 73.8 |

TT | 15.7 | 49.2 | 89.9 | 133.9 |

This table shows the average gold spending, separated per type: feathers or caducite vs. nuggets.

How to read?

- as above, the columns represent the nb of spins over the season
- the rows are the average gold that you will need to spend to get the 5 feather, 5 caducite and 125 nuggets (TT is the sum of both)

Example: with 25 spins, you will need an average of 15.7 Gold. At 10 spins, you will need nearly 134 on average.

As you can see, the nb of spins is important: for every day you miss, you miss 5 nuggets, which will need to be bought for 1G each. Some can be gotten on the roulette, but this is far from compensating the loss of not playing.

## Conclusion

The numbers surprised me, it was much worse than I expected: 70% of even the most loyal players will need to fork out some gold to get the staff!

If you do not play every single day, your odds go down dramatically (= the cost goes up).

## Addendum: with 5 days left, am I on a good track?

I have seen a ton of questions along the lines of "I currently have 95 nuggets, 4 feathers and 2 caducite. Will I need to pay gold?"

I won't go into the nuggets, because that should be obvious: you get 5 per day if you play and you might get 1 or 2 from the roulette. It's rather easy to see how much you will need (125 - current stock - 5xnb of days left)

The feathers and caducite are a little trickier. Here is my attempt, based on the assumption that you will get the daily roulette **another 5 times **(I will try to update this regularly based on how many days left we have...).

Table 1: probability of getting all the requirements without paying gold:

0 | 1 | 2 | 3 | 4 | 5 | |

0 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% |

1 | 0.0% | 0.0% | 0.0% | 0.0% | 0.2% | 0.7% |

2 | 0.0% | 0.0% | 0.0% | 0.3% | 2.4% | 5.8% |

3 | 0.0% | 0.0% | 0.3% | 3.5% | 14.2% | 26.3% |

4 | 0.0% | 0.2% | 2.4% | 14.2% | 42.2% | 67.2% |

5 | 0.0% | 0.7% | 5.8% | 26.3% | 67.2% | 100.0% |

The columns represent the stock of feathers you have today. The rows the nb of caducite you have today.

The percentage is the chance you have to get the 5 feathers and 5 caducite (at least) by the end of the tournament. So, for example, if your current stock is 5 and 3, you have 26% chance of getting all you need without paying gold. Note, for example, that you are better off with 4 and 4 (52%), although that is the same nb of cad/feathers in total.

Table 2: average gold needed

we have established your probability of needing gold. But how much?

0 | 1 | 2 | 3 | 4 | 5 | |

0 | 80.0 | 70.0 | 60.1 | 50.6 | 43.3 | 40.0 |

1 | 70.0 | 60.0 | 50.1 | 40.7 | 33.3 | 30.0 |

2 | 60.1 | 50.1 | 40.1 | 30.7 | 23.3 | 20.1 |

3 | 50.6 | 40.7 | 30.7 | 21.3 | 13.9 | 10.6 |

4 | 43.3 | 33.3 | 23.3 | 13.9 | 6.6 | 3.3 |

5 | 40.0 | 30.0 | 20.1 | 10.6 | 3.3 | 0.0 |

rows and columns, as above, represent your current stock of feathers / caducite. The cells give you the expected gold investment you will need based on your current position.

Again, if you have currently 5/3, you have 74% chance of needing gold. Sounds bad. But the second table tells us you will need to invest on average 10.6 gold. Still relatively affordable.

These are averages. It is clearly not possible to pay 10.6 G. What it means is that for players with 5/3, 26% will pay 0 (probability of getting 2 or more successes), 41% will need 10G (1 success) and about 33% will need 20G (0 success). On average: 10.6G.

## PS: methodology

for all the other stat-obsessed players out there, here is how I calculated the figures above:

- I first calculated a binomial distribution with a "success" probability of 40% (= getting either a feather or a caducite) for all nbs of spins (0 to 25)
- I then calculated the odds of the different combinations (=if I have 10 "successes" in the 1st round, what is the probability of getting 5 & 5, or 4 & 6...)
- Combining the 1st and the second table, I can get, for any nb of spin, the probability of all combos and the gold cost (=I know what the probability is of getting 6 feathers and 4 cads, and I know it will cost me 10G)
- for any point in the table (e.g, 20 spins, 12 "successes"), I can also calculate the probability of the number of nuggets (I have 5x the nb of spins + binomial calculation of 20-12 = 8 remaining spins with 50% probability)
- some macros and copy-paste later, I obtain the tables above.