Suppose I observe a highly improbable outcome while playing roulette - for example, 50 black results in a row, with a probability of 1/2^50. A mathematician would likely say the probability remains 1/2 for the next spin, since each event is independent.
From a physical or empirical perspective, however, if we see an outcome that is highly unlikely according to our mathematical model (like 50 black results in a row), it might be reasonable to suspect that there is something wrong with our model or with the device that is generating the outcomes (in this case, the roulette wheel). The physicist might argue that the device is biased in some way, or that some other physical factor (like a magnet) is influencing the results. And he might argue that the chance of black in next spin is 100%.
Now there a third way to answer this - if the wheel is broken, there is a chance it's not broken to give us blacks all the time. Maybe it's programmed to give white now, right when we are asked? In this case the correct answer to this - "Impossible to evaluate probability since we are unaware of the nature of the wheel's faulty behaviour (if we suppose there is one)?
As a philosopher seeking truth through reasoned inquiry, how should I evaluate this scenario? Should I trust the mathematically calculated 1/2 odds, or does the extremely improbable empirical result suggest the wheel is biased? Is it possible to integrate these perspectives?
More broadly, how do we address the tension between conceptual probability models and observations of real-world randomness that seem to defy the odds?