it is currently impossible to provide a precise statistical probability for the occurrence of the Big Bang without an external cause. Here are a couple of reasons why:

Incomplete Information: While our understanding of the universe and its origins has greatly improved thanks to scientific theories like the Big Bang theory and observations such as cosmic microwave background radiation, there's still much we don't know about the precise conditions preceding the Big Bang.

Limitations of Current Science: Theoretical physics, including quantum mechanics and general relativity, have provided some insights into the possible causes of the Big Bang, but these theories still have their limitations. For example, quantum mechanics and general relativity describe the laws of nature on different scales, and they haven't been fully reconciled into a single theory.

Unpredictability and Uncertainty: According to quantum mechanics, certain phenomena, like the quantum fluctuations that might have led to the Big Bang, are fundamentally probabilistic. This means that even if we had a complete understanding of the laws of physics, we would still only be able to talk about probabilities, not certainties.

Moreover, calculations of such probabilities require a theory that could describe the quantum behavior of gravity, and such a theory, often referred to as quantum gravity, is yet to be established.

So, while your question is interesting and touches upon deep and unsolved problems in theoretical physics, it's beyond the scope of our current scientific understanding to provide an estimate of the statistical probability of the Big Bang occurring without an external mover.
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