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Introduction

The last patchnotes revealed that the odds of getting a cover were influenced by a special mechanism designed to avoid too many duplicates.

This makes my previous blog post on the topic obsolete, as I had assumed that you could theoretically get an infinite amount of dups.

The new (actually old, but we did not know that) rules

The additional rule that we did not know about was:

"You have a chance to receive duplicate covers until you have received a number of duplicates equal to the number of unique covers you currently own..."

I understand it as such:

1st cover will never be a duplicate (duh!)

2nd cover: you will not have more than 1 duplicate (your amount of unique cover before getting the second one) > max 2 covers to get the 2nd unique, which is to say that you are sure to get the second cover with 30 boxes (10 for the 1st cover, max 20 for the second).

3rd cover: max 2 duplicates = max 3 covers to get it = maximum 60 boxes (10 for first + 20 for second + 30 for third)

...

8th cover: max 7 duplicates = max 8 covers to get the elusive last unique cover = max 360 boxes in total

Additionally, the notes say that "However, each duplicate you receive REDUCES the chance you will get a duplicate on your next lockbox opening".

That's more problematic, since there is no precise description of exactly how the chance of a duplicate is being reduced every time we get a dup. It should mean that, if I have 7 unique covers and get a dup, the next batch of 10 will no longer have 87.5% chance of dup but...something less?

For the purposes of this blog, I will ignore that part. I will still assume that the dup odds are unchanged until we hit the "cap" (as described above). As such, this makes my calculations pessimistic: the tables below should therefore be considered a worst-case-scenario.

Odds for each individual cover

a simple table to start for the odds for each cover separately.


cover 1 2 3 4 5 6 7 8
10 100% 88% 75% 63% 50% 38% 25% 13%
20 13% 19% 23% 25% 23% 19% 11%
30 0% 6% 9% 13% 15% 14% 10%
40 0% 0% 5% 6% 9% 11% 8%
50 0% 0% 0% 6% 6% 8% 7%
60 0% 0% 0% 0% 10% 6% 6%
70 0% 0% 0% 0% 0% 18% 6%
80 0% 0% 0% 0% 0% 0% 39%

How to read:

  • each row is a number of boxes (I assume you open by 10)
  • each column represents the nth cover
  • each cell shows the odds to get the nth cover by opening a certain number of boxes.

Example: for the 3rd cover, I have 75% chance to get it on my first attempt (10 boxes), 19% chance on my second attempt (20 boxes) and 6% of my 3rd attempt (30 boxes). 40 boxes and above have now a 0% probability, as per the "cap" explained in the previous paragraph.

Combined odds

Now a little more complex:


Nb of unique covers
Nb of boxes 1 2 3 4 5 6 7 8
10 100.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
20 0.0% 87.5% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
30 0.0% 12.5% 65.6% 0.0% 0.0% 0.0% 0.0% 0.0%
40 0.0% 0.0% 25.8% 41.0% 0.0% 0.0% 0.0% 0.0%
50 0.0% 0.0% 7.8% 31.5% 20.5% 0.0% 0.0% 0.0%
60 0.0% 0.0% 0.8% 16.7% 26.0% 7.7% 0.0% 0.0%
70 0.0% 0.0% 0.0% 8.0% 21.3% 14.6% 1.9% 0.0%
80 0.0% 0.0% 0.0% 2.2% 14.7% 17.1% 5.1% 0.2%
90 0.0% 0.0% 0.0% 0.5% 9.7% 16.2% 8.1% 0.8%
100 0.0% 0.0% 0.0% 0.0% 4.8% 13.8% 10.1% 1.8%
110 0.0% 0.0% 0.0% 0.0% 2.0% 11.6% 11.0% 2.8%
120 0.0% 0.0% 0.0% 0.0% 0.7% 8.3% 11.2% 3.8%
130 0.0% 0.0% 0.0% 0.0% 0.2% 5.2% 11.5% 4.7%
140 0.0% 0.0% 0.0% 0.0% 0.0% 2.9% 10.8% 5.6%
150 0.0% 0.0% 0.0% 0.0% 0.0% 1.5% 9.2% 6.9%
160 0.0% 0.0% 0.0% 0.0% 0.0% 0.7% 7.2% 8.3%
170 0.0% 0.0% 0.0% 0.0% 0.0% 0.2% 5.2% 9.2%
180 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 3.7% 9.4%
190 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 2.3% 9.0%
200 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 1.3% 8.2%
210 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.7% 7.5%
220 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.3% 6.4%
230 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 5.1%
240 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 3.8%
250 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 2.6%
260 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 1.8%
270 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 1.1%
280 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.6%
290 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.3%
300 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1%
310 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1%
320 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
330 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
340 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
350 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
360 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

as before:

  • each row is a number of boxes (I assume you open by 10)
  • each column represents the nth cover
  • each cell shows the odds to get the nth cover by opening a certain number of boxes.

The big difference is that we now look at the cumulated number of boxes needed: to get the 3rd cover you will also need the 1st and the second, so you will need from 30 boxes (if you get 3 unique covers) to 60 boxes (maximum bad luck).

So, for example, the odds of getting your 8th cover exactly with 230 boxes are 5.1%

This table is good for reading how many boxes you might need to reach a certain level of covers.

Combined cumulated odds

one step more:


Nb of unique covers
Nb of boxes 1 2 3 4 5 6 7 8
10 100.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
20 12.5% 87.5% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
30 0.0% 34.4% 65.6% 0.0% 0.0% 0.0% 0.0% 0.0%
40 0.0% 8.6% 50.4% 41.0% 0.0% 0.0% 0.0% 0.0%
50 0.0% 0.8% 26.7% 52.0% 20.5% 0.0% 0.0% 0.0%
60 0.0% 0.0% 10.8% 42.7% 38.8% 7.7% 0.0% 0.0%
70 0.0% 0.0% 2.8% 29.4% 45.6% 20.3% 1.9% 0.0%
80 0.0% 0.0% 0.5% 16.9% 43.2% 32.3% 6.8% 0.2%
90 0.0% 0.0% 0.0% 7.7% 36.7% 40.5% 14.0% 1.1%
100 0.0% 0.0% 0.0% 2.9% 27.8% 44.1% 22.4% 2.8%
110 0.0% 0.0% 0.0% 0.9% 18.1% 44.7% 30.6% 5.6%
120 0.0% 0.0% 0.0% 0.2% 10.5% 41.9% 38.0% 9.5%
130 0.0% 0.0% 0.0% 0.0% 5.5% 35.6% 44.7% 14.2%
140 0.0% 0.0% 0.0% 0.0% 2.6% 27.7% 50.0% 19.8%
150 0.0% 0.0% 0.0% 0.0% 1.0% 20.0% 52.3% 26.7%
160 0.0% 0.0% 0.0% 0.0% 0.4% 13.5% 51.1% 35.0%
170 0.0% 0.0% 0.0% 0.0% 0.1% 8.5% 47.2% 44.2%
180 0.0% 0.0% 0.0% 0.0% 0.0% 4.9% 41.5% 53.5%
190 0.0% 0.0% 0.0% 0.0% 0.0% 2.6% 34.9% 62.5%
200 0.0% 0.0% 0.0% 0.0% 0.0% 1.3% 28.0% 70.7%
210 0.0% 0.0% 0.0% 0.0% 0.0% 0.6% 21.3% 78.2%
220 0.0% 0.0% 0.0% 0.0% 0.0% 0.2% 15.2% 84.5%
230 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 10.3% 89.6%
240 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 6.6% 93.4%
250 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 4.0% 96.0%
260 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 2.2% 97.8%
270 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 1.2% 98.8%
280 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.5% 99.5%
290 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.2% 99.8%
300 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 99.9%
310 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%
320 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%
330 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%
340 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%
350 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%
360 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 100.0%

one difference to the previous one, this is cumulative. So the odds of having your 8th cover within 230 boxes are 89.6%.

this table is good for checking how many covers you are likely to have when you open a certain amount of boxes.

Conclusions

a few points to finish:

  • the average nb of boxes needed in this calculation is around 180, so about 30 less than in the previous calculations, and probably a little more in line with the experience of most players from the comments I had read on the wiki
  • this is a worst case scenario, as noted in the intro, so the real number should be lower
  • spending gold for that last elusive cover is now a lot more logical, since you know that you will at most spend 80G, and have roughly 60% chance of spending less (from the 1st table: 50% chance of spending 50G or less)
  • why the h... have they not told us sooner about this balance mechanism!? I, for one, might have spent gold on constrictor

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