Problem Regarding Buoyancy 
A spherical marble of radius $1\,$cm is stuck in a circular hole slightly smaller than its own radius (for calculation purposes , both are equal) at the bottom of a filled bucket of height $10\,$cm. Find the force on the marble due to the water.

I have always been troubled by problems like this. Does the marble not displace a certain volume of fluid? Should a buoyant force not act on it? However, in this problem, the answer happens to equal the product of the pressure, and the projection area....
And, when I came across this similar problem :- 

A steel ball is floating in a trough of mercury. If we fill the empty part of the trough with water, what happens to the steel ball?

The answer to this one is that the steel ball rises. 
Here, instead of multiplying the pressure and area of projection, and arguing that a net downward force acts, we argue that the steel ball displaces water, and causes an upward buoyant force to act.
My question is, when does one know which force to apply?
 A: It all depends on the fluid contact.
Buoyancy comes about due to hydrostatic pressure differences on a submerged or floating object.
For submerged or floating objects, the fluid pressure acts on the submerged volume.  Because fluid pressure increases with depth due to hydrostatics; when you submerge an object, the pressure at the top is less than the pressure at the bottom.  This causes the net upwards force on the object that we call buoyancy.  As long as the fluid is below the object, it will have buoyant force.
When your marble is in the bottom of the bucket with only fluid above it, it is not the same as being submerged.  There is no higher pressure fluid below it, only a high pressure fluid above it, so the net force due to fluid acts downwards, not upwards as it does when the top and bottom faces of the object have pressure acting on them.
TL;DR: You need to see if the fluid is actually surrounding the ball as in the second case, or if it's just acting on top of the ball, as in the first case.
A: For the first problem, you should know that buoyancy arises due to pressure difference which in turn arises due to mass of fluid above a certain level.
You are correct in saying that the ball will displace a volume of water, and that is equal to half the volume of the sphere.
So we can write that the force acting due to water will be due to the (mass of water above the ball)g, which is just another way of saying that it is equal to pressure times projection area,
and the actual volume of water above the ball = (h*Area of base of cylindrical region) - (Half the volume of sphere)
And now it's quite easy to proceed.
