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The simple answer would seem to be 25%:
- 8 inner cubes with no sides painted: 0%
- 24 face cubes with one side painted: (24 ÷ 64) × (1 ÷ 6) = 6.25%
- 24 edge cubes with two sides painted: (24 ÷ 64) × (2 ÷ 6) = 12.5%
- 8 corner cubes with three sides painted: (8 ÷ 64) × (3 ÷ 6) = 6.25%
- 6.25% + 6.25% + 12.5% = 25%
EDIT: No, a cube has six sides.  
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
- Homer
modified 9-Feb-17 4:45am.
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Richard, a cube has 6 sides.
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And what if you pick one of the inner cubes without red?
You forgot to add the probability of picking one random out of the box.
M.D.V. 
If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about?
Help me to understand what I'm saying, and I'll explain it better to you
Rating helpful answers is nice, but saying thanks can be even nicer.
modified 9-Feb-17 5:26am.
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Nelek wrote: And what if you pick one of the inner cubes without red?
That's the first line: 8 inner cubes with no sides painted. Since there are no sides painted, the probability of the cube landing with a red side face-up given that you've picked one of these is 0%.
Nelek wrote: You forgot to add the probability of picking one random out of the box.
No I didn't. I might have initially cocked-up the number of sides a cube has, but I didn't forget "picking one at random" part. That's what the (24 ÷ 64) and (8 ÷ 64) parts are about.
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
- Homer
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My bad... lack of coffee.
Somehow I thought that 25% was still too high and after the error of the faces of a cube... I thought you were missing coffee too 
M.D.V.
If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about?
Help me to understand what I'm saying, and I'll explain it better to you
Rating helpful answers is nice, but saying thanks can be even nicer.
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That's the first correct answer, with an explanation. But it's not the simplest. 
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Jörgen Andersson wrote: That's the first correct answer, with an explanation.
But only after @CPallini reminded me how many sides a cube has. 
And technically, his answer[^] was the first correct answer. He just forgot to explain it.
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
- Homer
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Richard Deeming wrote: He just forgot to explain it
He did indeed, so that's no win.
But since he gave you some important help I suppose you should share the honours.
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No explanation was required.  However I used Richard's approach.
Now I see the simpler one (surface vs volume): The little red faces can be computed on the big red cube: 4x4x6. The odds of having a red face should be wighted by the total little faces, that is how many little cubes there are multiplied by 6: (4x4x4x6), hence 1/4.
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CPallini wrote: No explanation was required
That's Ok. There's no price either.
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Winning is priceless.
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There are 64 1" cubes total. The outside is 16 1" cubes times 6 sides. The total is 64 1" cubes with 6 sides. (16 x 6)/(64 x 6) = 1/4.
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For all answers doing the statistics (mathematics):
Don't forget the physics. Cubes with one or more painted sides are not perfect cubes and chances that a painted side is at the bottom are slightly higher than to be on the top
Oh, and when the paint is not dry yet it is even more improbable that a red face is on top.
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You damned cheating gambler!
However, you are assuming the paint density greater than cube's one. 
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Even with the same density the weight and the thickness of the painting make the cube unbalanced and not being a perfect cube.
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That's true. However you cannot take for granted the disvantage of having the red face up.
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Jochen Arndt wrote: Don't forget the physics.
OK: "Assume a spherical cow of uniform density, while ignoring the effects of gravity, in a vacuum." 
Abstruse Goose | Spherical Cow[^]
Spherical cow - Wikipedia[^]
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
- Homer
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Richard Deeming wrote: while ignoring the effects of gravity, in a vacuum
Then the solution is simple:
0% because the cube will never stop rolling
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31,25 % ?
33,3 % ?
I finally saw that i had a small error...
1*0,5
3*33,3%
3*16,6%
1*0%
gives 2/8 instead of 2,5/8 XD
so yeah 25% (Also if you check the sides like 96R/384A)
Rules for the FOSW ![ ^]
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modified 9-Feb-17 5:29am.
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If my thinking didn't fail me there is 25% probability ... that said, the way I see it, the problem can be reduced to a variation of picking the right color ball from the bag, with the "ball" being right color (1˝) cube side, therefore we have 96 colored balls cube sides and 384 all cube sides.
modified 19-Nov-18 21:01pm.
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This!
Not the first correct answer, but the best!
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Personally, I think that discussions about ball bags belong in the soapbox.
I wanna be a eunuchs developer! Pass me a bread knife!
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Zero*
*because you didn't let the paint dry therefore it is sticky
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There is a 37.23568 % chance of the cube asking for more bacon?
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