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So if you are one of those guys asking why you haven't gotten 8 different comic covers and thereby Magneto, then this is the place to look. At first I'll only bring the probabilities for the distribution, after opening 8 lockboxes, but maybe I'll do more calculations later. I'll only do calculations on probabilities when you open 10 boxes at a time since that is the most effective way of getting the covers. This means that 8 openings is equal to 80 boxes.

Notation

In this blog, I'm going to calculate the probability of the number of owned covers after a number of openings. I'm using the letter P for probability. I'm writing it as P_n(x) where x is the number of covers after n openings. This means that x will always be between 1 and 8.

For instance

P_3(1)=\tfrac{1}{64}\cdot 100\ \%= 1.56\ \%

means that the probability of having only one cover after three openings are as small as 1.56 %.

The "grid"

19nkNrF.jpg
The numbers are the number of different covers owned, and the fractions are the probabilities for the next

The first time you open boxes, you obviously don't have any covers yet and therefore have 100 % chance of getting a new cover or a chance of \tfrac{8}{8}.

The second time you open boxes you can either get the same cover again \left(\tfrac{1}{8}\right) or get a new one \left(\tfrac{7}{8}\right), so now we have a branching.

After the third opening you can two \left(\tfrac{7}{8}\cdot\tfrac{2}{8}\right) or three \left(\tfrac{7}{8}\cdot\tfrac{6}{8}\right) covers if you already had two different and you can have one \left(\tfrac{1}{8}\cdot\tfrac{1}{8}\right) or two \left(\tfrac{1}{8}\cdot\tfrac{7}{8}\right) different if you got the same one twice.

Now observe that there are two ways of having two covers after three openings and therefore in order to find the probability of having 2, we need to sum up the probabilities for each of the two ways. This means that

P_3(2)=\tfrac{7}{8}\cdot\tfrac{2}{8}+\tfrac{1}{8}\cdot\tfrac{7}{8}=\tfrac{14}{64}+\tfrac{7}{64}=\tfrac{21}{64}=0.3281=32.81\ \%

The probability for having 3 or 1 album after 3 openings is somewhat simpler, since there will be only one way of getting each of them. The probabilities are

P_3(1)=1.56\ \%

and

P_3(3)=65.63\ \%

Now in order to check if we calculated correct the sum of all probabilities should give 100 %:

P_3(1)+P_3(2)+P_3(3)=1.56\ \% + 32.81\ \% + 65.63\ % = 100\ \%

This shows that in the start, it is quite easy to get a new cover each time, and the calculations were also relatively simple. We had a total of 4 ways and two of them led to two covers, one led to one cover, and the last one led to three covers. That makes perfectly sense since we would certainly not expect to get four different covers after only three openings, right?


But this is where the trouble begin, because when we are going to open boxes for the fourth time for each way we can go two more ways and now suddenly we can go 8 ways. For the fifth opening there are 16 different combinations and so on. This can be seen on the illustration of "The Grid" to the right.


How many covers do I have after 8 openings?

Here is when the interesting part begins. In this section we finally come to the part where we calculate the probabilities after 8 openings. If the interest is here, I will expand this section with the formulas used for the calculations, but for now I will just postulate my results in the table below. When time is the more relevant question, how many boxes do I need to open to get Magneto? will get answered.


Number of covers after 8 openings Probability
1 0.00 %
2 0.04 %
3 1.93 %
4 17.03 %
5 42.06 %
6 31.96 %
7 6.73 %
8 0.24 %

So it is actually more difficult to get Magneto after only 8 tries than it is to win a PvP hero. That is if we assume that every player has the same conditions in PvP which is obviously very incorrect. Jokes aside as you can see a little more than every third player gets at least 6 covers after the first 8 openings and more than 80 % get at least 5 covers. So the conclusion is that you should not expect to get Magneto after you have opened your first 80 boxes. And remember: when you have 7 covers, no matter how many boxes you have opened, there will always be 12.5 % chance to get the final one each time you open another set of ten boxes!

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