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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 ![ ^]
if(this.signature != "")
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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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What's wrong with the other cubes?
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The other cubes are vegans. 
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I need a business case and full specifications, before I can commit to this project.
I wanna be a eunuchs developer! Pass me a bread knife!
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It depends on which of the 1-inch cubes you select.
8 will have 3 painted sides (1 in 2)
24 will have 2 painted sides (1 in 3)
24 will have 1 painted side (1 in 6)
8 will have no painted sides (0 chance of getting a painted side up)
Which cube is selected?
".45 ACP - because shooting twice is just silly" - JSOP, 2010
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You can never have too much ammo - unless you're swimming, or on fire. - JSOP, 2010
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When you pry the gun from my cold dead hands, be careful - the barrel will be very hot. - JSOP, 2013
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Jörgen Andersson wrote: You have a 4” cube, and spray the outside with red paint.You cut this big red cube into individual 1” cubes (the freshly exposed faces do not have any red paint on them).All of these little cubes are placed into an urn, are thoroughly mixed, and one selected at random.This chosen cube is then rolled like a die.What is the probability that it lands with a red face up?
I know that if had the inclination to do this, my probability of getting laid would be 0.
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100%. You didnt wait for the paint to dry. 
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16*6/16*4*6 = 1/4
number of painted faces (16 on each of 6 sides) / total number of faces (16 cubes per layer, 4 layers, and 6 sides per cube).
Did you ever see history portrayed as an old man with a wise brow and pulseless heart, waging all things in the balance of reason?
Is not rather the genius of history like an eternal, imploring maiden, full of fire, with a burning heart and flaming soul, humanly warm and humanly beautiful?
--Zachris Topelius
Training a telescope on one’s own belly button will only reveal lint. You like that? You go right on staring at it. I prefer looking at galaxies.
-- Sarah Hoyt
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100%. My cube is made from red plastic, so the face color is immaterial.
Software Zen: delete this;
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