## Introduction

One of the users (Wearingglasses) wrote a nice blog post during Ops 6 about how many boxes you needed to open to have a good chance of getting your 8 covers.

He did a simulation and concluded that you needed 220 boxes to start being confident (60% + probability) to get the 8 covers. you can find his blog here

The comments section of the blog above already showed some links where you could find calculation methods, but I went a little further and calculated the whole thing...now you can finally put to bed the question being asked time and again: I got 5 dupes from 110 boxes, is it lucky or unlucky??

Update: as far as I can see, the maths for OS still seems valid for Juggernaut.

## Probability table:

How to read the monster below? Simple. Each row represents the nb of 10-box-lots you opened, and you can read in the column the probability of getting a certain nb of unique covers.

Example: if you open 60 boxes (row 6), you have a 41.7% chance of getting 4 unique covers (4th column)

Nb of unique covers | ||||||||

Nb of boxes | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

10 | 100.0% | |||||||

20 | 12.5% | 87.5% | ||||||

30 | 1.6% | 32.8% | 65.6% | |||||

40 | 0.2% | 9.6% | 49.2% | 41.0% | ||||

50 | 0.0% | 2.6% | 25.6% | 51.3% | 20.5% | |||

60 | 0.0% | 0.7% | 11.5% | 41.7% | 38.5% | 7.7% | ||

70 | 0.0% | 0.2% | 4.8% | 28.0% | 44.9% | 20.2% | 1.9% | |

80 | 0.0% | 0.0% | 1.9% | 17.0% | 42.1% | 32.0% | 6.7% | 0.2% |

90 | 0.0% | 0.0% | 0.8% | 9.7% | 34.8% | 39.7% | 13.9% | 1.1% |

100 | 0.0% | 0.0% | 0.3% | 5.3% | 26.6% | 42.9% | 22.1% | 2.8% |

110 | 0.0% | 0.0% | 0.1% | 2.9% | 19.3% | 42.1% | 30.0% | 5.6% |

120 | 0.0% | 0.0% | 0.0% | 1.5% | 13.5% | 38.8% | 36.8% | 9.3% |

130 | 0.0% | 0.0% | 0.0% | 0.8% | 9.2% | 34.2% | 41.9% | 13.9% |

140 | 0.0% | 0.0% | 0.0% | 0.4% | 6.1% | 29.1% | 45.2% | 19.2% |

150 | 0.0% | 0.0% | 0.0% | 0.2% | 4.0% | 24.1% | 46.8% | 24.8% |

160 | 0.0% | 0.0% | 0.0% | 0.1% | 2.6% | 19.6% | 47.0% | 30.7% |

170 | 0.0% | 0.0% | 0.0% | 0.1% | 1.7% | 15.7% | 46.0% | 36.6% |

180 | 0.0% | 0.0% | 0.0% | 0.0% | 1.1% | 12.4% | 44.2% | 42.3% |

190 | 0.0% | 0.0% | 0.0% | 0.0% | 0.7% | 9.7% | 41.8% | 47.8% |

200 | 0.0% | 0.0% | 0.0% | 0.0% | 0.4% | 7.5% | 39.0% | 53.1% |

210 | 0.0% | 0.0% | 0.0% | 0.0% | 0.3% | 5.8% | 36.0% | 57.9% |

220 | 0.0% | 0.0% | 0.0% | 0.0% | 0.2% | 4.5% | 32.9% | 62.4% |

230 | 0.0% | 0.0% | 0.0% | 0.0% | 0.1% | 3.4% | 29.9% | 66.5% |

240 | 0.0% | 0.0% | 0.0% | 0.0% | 0.1% | 2.6% | 27.0% | 70.3% |

250 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 2.0% | 24.3% | 73.7% |

260 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 1.5% | 21.8% | 76.7% |

270 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 1.1% | 19.4% | 79.4% |

280 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.9% | 17.3% | 81.9% |

290 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.6% | 15.3% | 84.0% |

300 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.5% | 13.6% | 85.9% |

310 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.4% | 12.0% | 87.6% |

320 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.3% | 10.6% | 89.1% |

330 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.2% | 9.3% | 90.5% |

340 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.2% | 8.2% | 91.6% |

350 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.1% | 7.2% | 92.6% |

360 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.1% | 6.4% | 93.6% |

370 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.1% | 5.6% | 94.3% |

380 | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 4.9% | 95.0% |

Quick conclusions from the table

- at 220 boxes, you are at 62% chance of having the 8 covers, confirming the simulations from Wearingglasses
- you go over 50% probability at 200 boxes
- you need 380 boxes to reach the 95% probability (= near certain) to get the 8 covers.
- I was lucky to get Magneto with 170 boxes only (37% chance)

## How Many boxes more will I need to open?

Another interesting question: I have X covers already, how many boxes more will I probably need to open?

Of course, if you got lucky and got 7 unique covers from your first 70 boxes, you most likely will not need to open 220 boxes, you now have a good chance at getting there faster. But how fast?

Here is another obscure table, from your starting nb of unique covers, how many more boxes you need:

50% | 75% | 95% | |

1 | 190 | 250 | 370 |

2 | 180 | 240 | 360 |

3 | 160 | 220 | 350 |

4 | 150 | 210 | 330 |

5 | 130 | 190 | 310 |

6 | 100 | 160 | 280 |

7 | 60 | 110 | 230 |

How to read that one: the row is determined by how many covers you have now (regardless of how many boxes you opened so far). Let's say 6.

The 1st column shows you how many boxes you will need to open to get over 50% probability of having the 8 unique covers, in this case, 100 more.

So, if you already have 7 (my case), you will most likely need to open more than 60 additional boxes, and probably less than 110.

## Why it is better to open boxes 10 at a time

Addendum to the original post: the argument keeps popping up that it might be better to open the boxes 1 by 1. The logic being that, while you could get 0 covers in 10 boxes, you could also get 2 or more, thereby increasing your chances.

here is my calculation:

Nb of covers | |||||

Nb of boxes | 0 | 1 | 2 | 3 | 4 |

1 | 93.0% | 7.0% | 0.0% | 0.0% | 0.0% |

2 | 86.5% | 13.0% | 0.5% | 0.0% | 0.0% |

3 | 80.4% | 18.2% | 1.4% | 0.0% | 0.0% |

4 | 74.8% | 22.5% | 2.5% | 0.1% | 0.0% |

5 | 69.6% | 26.2% | 3.9% | 0.3% | 0.0% |

6 | 64.7% | 29.2% | 5.5% | 0.6% | 0.0% |

7 | 60.2% | 31.7% | 7.2% | 0.9% | 0.1% |

8 | 56.0% | 33.7% | 8.9% | 1.3% | 0.1% |

9 | 52.0% | 35.3% | 10.6% | 1.9% | 0.2% |

10 | 48.4% | 36.4% | 12.3% | 2.5% | 0.3% |

Each row corresponds to the number of individual boxes you open. The columns to the numbers of covers you get (not necessarily unique covers, just covers). The table shows the probability of any combination. For example: if you open 7 boxes, you have only 31.7% chance of having exactly 1 cover but 60.2% chance of having none.

For 10 boxes, the conclusion is that:

- in 36% of cases, you will get 1 cover (same as if you opened them by 10)
- in a little over 15% of cases you will get 2 or more covers (but most likely 2)
- in 48% of cases (!), you will get 0

In other words: you are 3 times more likely to get less covers than you are to get more.

I don't know about you, but I do not like those odds.

## Assumptions / caveats

As with all the calculations I have seen so far, I assumed that the probability of each cover is exactly equal for every batch of 10 boxes you open. I do not know that, but I have not seen/heard of evidence to the contrary.

All calculations in the first 2 tables also assume that you open boxes by batches of 10, not individually. If you open individually, over a large number of boxes, you are near certain to get less covers and therefore decrease your chances of getting Omega-Sentinel (see last section).

I am not a mathematician or a statistician, so I might be wrong: feel free to give your input!

Good luck to all of you with the Omega-Lockboxes!