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Well, assuming it is in a random state, let’s do the math.

There are 9 center pieces on each center.
Each one can be one of each center. The center one can be discounted because it can be the base center. The first corner one we do can be determined like this:
4 corner pieces per center.
6 colors. 6x4=24. This means that there are 24 possible pieces that could be there, and 4 of them are right, leaving a 1/6 chance it will be right. The next corner has 3/24 or 1/8 because there is one less correct center. Then 1/12 and finally 1/24. Multiply these and you get 1/13824 that all corners and the center are right. Then for edges, you can do the same thing because there are the same number as corners. So 1/13824 squared gets you 1/191102976 chance that a particular center will be solved, but for any of the 6 centers that goes up to:

1/31850496 chance that a center on the cube will be solved.

For yellow/white it would have to be 3X as unlikely, 1/95551488, because yellow/white make up a third of the 6 centers.

These numbers seem too high, but I think I did everything right, so would someone check it?

maybe that is for just one center you calculated so it is 1/5308416
also I think it should be 3/23 for the second one,2/22 for the third and so on since you are eliminating one piece

chances of getting a ll skip no auf, combined with cross being a one mover, combined with all the pairs except one inserted already in f2l?

ok that was a joke but seriously, im thinking is there a difference in probability with ll skips having auf, or no auf. I think that the no auf ones are more rare.

read the gods number article on ruwix.com "This number may seem low, but theoretically it should be even lower. Only around 490,000,000 combinations require the full 20 moves to be solved. Although 490 million is a huge number, it is only a fraction of the 43 quintillion possible combinations (0.0000011328955% to be precise). " so .0000011 percent of positions take 20 moves to solve

read the gods number article on ruwix.com "This number may seem low, but theoretically it should be even lower. Only around 490,000,000 combinations require the full 20 moves to be solved. Although 490 million is a huge number, it is only a fraction of the 43 quintillion possible combinations (0.0000011328955% to be precise). " so .0000011 percent of positions take 20 moves to solve

Not bad, if you think about it. The chances of being struck by lightning is 1 in a million, which means that the chance of the 20 move scramble would be less rarer.

read the gods number article on ruwix.com "This number may seem low, but theoretically it should be even lower. Only around 490,000,000 combinations require the full 20 moves to be solved. Although 490 million is a huge number, it is only a fraction of the 43 quintillion possible combinations (0.0000011328955% to be precise). " so .0000011 percent of positions take 20 moves to solve

read the gods number article on ruwix.com "This number may seem low, but theoretically it should be even lower. Only around 490,000,000 combinations require the full 20 moves to be solved. Although 490 million is a huge number, it is only a fraction of the 43 quintillion possible combinations (0.0000011328955% to be precise). " so .0000011 percent of positions take 20 moves to solve