There are two optical phenomena which are happening to make the sky appear blue.
The primary on is due to refraction:
As you know the sun bathes the Earth in pretty close to white light (a bit heavy on the yellow side). The light is generally described by the "colors of the rainbow": red, orange, yellow, green, blue, indigo, and violet. As the light leaves the sun and gets to the Earth all of the colors are mixed in the rays of light.
When the rays strike the Earth's atmosphere they are entering a more optically dense medium. According to the law of refraction, when a ray of light goes from less dense to more dense medium the rays are bent. It is also true that different wavelengths of light are bent at different angles. The blue rays of light are bent at just the right amount that we can see them with our eyes. Other colors are bent too sharply and we don't see them, or not bent enough and we don't see them. We do get to see them if we are looking out over a flat area toward the horizon. As the sun rises and falls we can see each of the colors.
The second effect is due to light absorption and scattering.
Molecules carried in the atmosphere (primarily water) absorb and scatter light. Blue wavelengths of light are absorbed less and scattered more and so that color is also projected toward the earth's surface more.
The sky is not grey because we are not mixing blue with black.The laws for mixing paint(subtractive) are not the same as mxing light(additive).
In the sky all colors of the spectrum are bent by the atmosphere. Blue is bent just enough to make it the predominant color most of the day but not at suntise or sunset.Simply put the red,orange,yellow,green,and violet are bent and the intense blue is seen.
There is no black mixed in and the black of outer-space is not the same phenomenon as black paint or ink. Black paint and ink absorbs most of the light reflected off it.When you mix black with blue paint ,the particles of black absorb some of the blue light that the blue paint particles would normally reflect .This results in a muted,greyed blue. The blue light in the atmosphere acts totally as though it were in front of outerspace.
Think of mixing blue and some black paint together.You'll get a greyish tint.However paint something black let it dry.Then paint blue on top of it,the layer of pigment closest to your eye if put on with enough thickness is completely going to ''rule''. This would be the best analogy as all the blue is in ''front'' of the blackness and not the same as a mix of black and blue paint.