I am a mathematician who has recently started delving into philosophy and I had a thought today. From what I understand, the scientific method can be described in the following terms.
Suppose we have an experiment E to run with pre-conditions P to be assured are in a certain state, then after the pre-conditions are set, we observe an outcome O. Then for each additional experiment E' with the same pre-conditions with the same outcome O, the estimated probability of the causal relationship between the pre-conditions and the outcome O increases. By the law of large numbers the sample mean should converge to the true mean, where if we assign 0 to outcome O not occuring, and 1 to outcome O occurring, if causation is consistent and unwavering as in a physical law, then the sample mean should equal to the true mean, which is 1.
Then my question is: how could one distinguish between a world in which the probability of a causation in some scenarios not occurring is non-zero. For example if the probability that after I turn on the heat on my stove to the boiling temperature of water and put a pot of water on, the probability the water will not boil is non-zero in some world A, albeit very very small, and is zero in some world B. Then presumably in world A, strange things should occur a non-zero number of times, but catching a rare probabilistic event in the act so to speak may never occur in the lifetime of humanity.
Again I am not a trained philosopher so maybe my question has false assumptions/problems, but any insights into this much appreciated.