Assuming a light source, emitting white light,a red colored diffuse sphere lit up by the light, and a white plane below the sphere acting as the 'floor', we find the diffuse reflection of the sphere on the floor to be reddish in color, just like the sphere itself

Why does it have the exact same color as the sphere? The emitted light was white in color, so when the photons of the light source, collide and reflect from the sphere, should they not still be white, causing the diffuse reflection to instead be white? What causes the photons to take on a reddish hue?

In the case, of a black sphere as opposed to a red sphere, can we expect the same thing to happen? Will the diffuse reflection be grey-black in color as well? If, on the spot where the reflection is cast,a light grey soft shadow, lighter than the diffuse reflection were to exist, would the reflection of the sphere, 'overpower' that of the shadow, making the region darker as a result?