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Maybe a more difficult and related probability question: If an artist/advertiser/etc. puts an image of a cube in their work, only showing three faces, what is the probability that there exists a possible sticker arrangement for the other three faces where it is solvable? Would be interesting to see this for when forcing the standard color scheme and when not forcing it.

The tricky part for me is that I'm not sure how to make a comprehensive list of the ways that stickering 3 sides can fail, gaining a list of restrictions on coloring the stickers, and then turning that into a counting problem.

Maybe a more difficult and related probability question: If an artist/advertiser/etc. puts an image of a cube in their work, only showing three faces, what is the probability that there exists a possible sticker arrangement for the other three faces where it is solvable? Would be interesting to see this for when forcing the standard color scheme and when not forcing it.

The tricky part for me is that I'm not sure how to make a comprehensive list of the ways that stickering 3 sides can fail, gaining a list of restrictions on coloring the stickers, and then turning that into a counting problem.

I'm no expert and I haven't given this as much thought as you, but my main issue with solving this problem stems from figuring out which questions not to answer.
My take on the problem, correct me if I'm wrong, before asking what the probability is that there exists a possible sticker arrangement for the other three faces where it is solvable, ask: "How do we know that the visible 3 faces indicate an unsolvable position?"

Obviously, 27 stickers must be visible.
There may not be more than 9 of a color visible in total.
There may not be more than one of a color on any center.
There may not be more than 4 of the same color on any corner or edge.
On any one piece, there may not be more than one of a specific color.

That's all I've got so far. With three invisible pieces, it's impossible to tell if there is illegal orientation parity or permutation parity in the edges or corners.

This is not the correct answer... the 54 stickers aren't all unique, so there are fewer than 54! ways to arrange them.

Once completely solved, this is your answer:
0.
And I mean this legitimately.
The computing power of the entire fleet of an average Airline would not be able to compute the answer to a quantifiable number other than 0.

No... the number is positive, which means it is not zero, and I can easily compute it to a lot of digits: 0.000000000000000000000000000000000000000000001422103989345743645216458545800499169872628252719370729936591517132547683030602326252059492655156448357673450533196070973736641912895463250288389528076468188229280281803031898896...

You don't take identical stickers into account, or different color schemes (the cube is still solvable with a weird color scheme - in the sense that you can turn the layers to get every face to have a solid color). Also, there's nothing "astonishing" about your result... and it is very easy to divide one integer by a larger one, you can do it by hand with paper and a pencil, or on any computer with the right software (even a programmable calculator or cheap smartphone).

If you're not gonna explain an equation in any way, you shouldn't bother writing it down.
What if your teacher explained permutations to you by just saying:
"We know the that we can arrange 6 elements in 720 different ways because 6!"

EminentCuber said:

If we take this number and divide it by the possible number of sticker arrangements: 54!. We can get the number the probability that if you randomly remove and put adhere stickers it will be a solvable permutation.