A well-defined quantum probability in the beginning of the universe? In mathematics or statistics, a well defined probability requires a large sample space.
However, in the beginning of the universe, when the first quantum collapse happened, the sample space contains only one event itself, you can not predict the probability of this event.
Only after billions of years, such collapse happened many times, we have a big sample space, thus a well-defined probability.
 A: @JohnRennie is correct.
Whenever one hits on a paradox one has to look at the assumptions that create it. Usually there is a confusion of two frameworks, each valid in its own region but the mixture creates a paradox. A paradox can also signal the need for a new framework, as happened with the ultraviolet catastrophe predicted by classical black body radiation which was resolved with the introduction of the quantum mechanical framework.
In the very title of the question, the confusion of two different frames is evident:

A well defined quantum probability in the beginning of the universe?

The beginning of the universe has a meaning only in classical General Relativity models.
Quantum probability has a meaning in quantized systems.
At the moment there is no  theory of everything (TOE) that both quantizes gravity and is physically meaningful by incorporating the standard model of particle physics. The need for any quantization of gravity to embed the SM comes because the SM encapsulates all we know about the physical world at the level of the laboratory.
One can hand wave and "solve" the paradox by saying : if gravity is quantized, the Heisenberg Uncertainty Principle , HUP, ensures that there is no unique value even at t=0, and there will be a quantum mechanical probability distribution even at that time for the state of energy and time and the state of momentum and space. Unless we have a TOE nothing is rigorous in these types of argument.
For example, in the classical electromagnetic problem of an electron around a proton there is a singularity, the potential becomes infinite at radius=0. Hand waving with the HUP we could say there is not singularity because the electron can be anywhere within the HUP instead of on the singularity. The real solution  of the quantum mechanical problem though is much more interesting: There are energy levels on which the electron can be trapped and stay there ad infinitum.
Thus, maybe if we nail the TOE ( and string theorists are trying hard) the reason that there is no singularity might be much more interesting than just the HUP argument.
Edit after community brought the question up again
Rereading you question I see a more basic confusion, that between experiment and theory.

However, in the beginning of the universe, when the first quantum collapse happened, the sample space contains only one event itself, you can not predict the probability of this event.

This can be seen as one experimental measurement/event, yes, even though the theoretical wavefunction predicting its probability  may be quite complex. Our universe is one experimental point to be added up to build an experimental  probability  distribution.

Only after billions of years, such collapse happened many times, we have a big sample space, thus a well-defined probability.

The probability function is well defined mathematically. It is a theoretical distribution.
If the experiment could be done, it would be an experimental validation of the probability function predicted by theory. We are stuck with our unique universe as one point to be added to an impossible experiment.  Mathematical theories can predict probability distributions without recourse to experiment.  In this particular case the quantum mechanical experiment cannot be carried out many times to have a validation of the specific probability function of the theory. BUT the theory predicts  other observations that validate it, for example the distribution of the Cosmic Microwave Background, and recently the BICEP2 results ( if they hold up with further analysis and observation from other experiments). Thus we consider the theory valdated up to now.
Keep in mind that a theory is never proven by experiment/observation, it is validated if it agrees with experiment. If it disagrees even with one instance of experiment/observation it  is falsified and has to be sent back to the drawing board.At the moment the inflation hypothesis is validated.
