Probability must be quantized, as everything in a finite universe must be quantized.
(And the universe must be finite or the finite elements would be meaningless.)
The probability of past events has been set at 100% - they were 100% at some present time, and cannot later be affected.
The probability of future events is always less, and considering the extreme complexity of the universe, the probability of any future state of the universe must be extremely small. With something like 1080 atoms in the universe, the chance of any particular state is essentially infinitesimal.
Because of the NIP, that means the probability of any universe state for any point in time more than epsilon (probably a Planck Time or thereabouts) in the future is exactly zero.
As a result, the past and future have a basic asymmetry - the past can be physical, but cannot be affected, (the probability of alternatives is fixed at 0%) while the probability of anything beyond the very near future is also 0%, but can increase. It is only the very next moment - with finite non-unity probabilities - that is, that can be, physically relevant.
Perhaps, then, the present is actually a part of the past, and the future is divided into two types.
I can see how this would be confusing. How is it that I'm not simply restating the existence of the arrow of time?
Alternatively, time could not be the independent variable, and instead could be basically a special type (it's an imaginary number in GR) of dimension. As a result, the probabilities would be 100% in both directions. If the arrow of time was truly arbitrary, or decided by the direction of increasing entropy then the probability asymmetry would not exist. (Or perhaps does not exist!)