How impossible is an event with zero probability?

When we look at reality, we always perceive it as extremely complex and hard to model. Modelling reality in a perfect way would essentially mean being able to explain and predict phenomena with no more need of probability. In general, probability is what leads our lives to a choice instead of another, preferring A over B on the base of some subjective and/or objective scoring.

The extreme complexity can be seen as a wide range of possibilities that may occur after some initial event (say, I leave home), each possibility having a very small probability to be observed. Of course, some is much more likely than the other (say, I come across my neighbour), maybe quite certain, but still there is a continuous chain of events that are not predictable (what will my neighbour do and/or say to me? What will I answer? Will I be late at work because of something happening on my way to the office?).

This long sequence of events that portrays our life has zero probability to happen. But still, it happens. This means that everything we observe is the result of a chain of events having zero probability. But every possible chain of events is almost impossible to be observed. Put it like this: if each possible sequence has zero probability, this means that all sequences have almost the same (very small) probability to occur. However, one of them has to occur for sure.

Handling complexity in a proper way means being able to build models that can explain some systematic pattern in reality, making some sequence much more likely to happen given certain inputs. Statisticians’ work is to unearth the deterministic in a world largely ruled by randomness. If the work is accomplished properly, then just the mere case remains unexplained.

Could we do something more? Well, it depends on what randomness really is. We may see it as a binder of what is not possible to observe for technical or natural reasons. For instance, if we knew the starting face of a coin, our mechanic movement when tossing it, the effect of the wind, etc…, of course we would be able to predict better than usual whether it is going to come up head or tail. But sometimes, it is just too difficult, costly, or just impossible to have a model that improves our predictions. This is why we end up gambling on the real unknown with an irrational certainty of being able to handle it.

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