The Mad Scientist Problem

The Problem

Let's imagine we have a mad scientist who is performing an experiment on a caged man who is unaware the mad scientist exists. In order to get food, the man must press a button on a food dispenser. However, the dispenser is rigged to deliver a mild electric shock to the man 50% of the time. After some time, the man learns that there is about a 50% chance of shock. At this point, the mad scientist then alters the machine to deliver a shock 75% of the time without the man's knowledge. This problem has a two questions:

1. Without further button pressing, does the man "know" there is a 50% chance of shock despite the scientist's alteration?
2. After encountering higher levels of shock, when should the man move from attributing the higher rate to a streak of bad luck to "knowing" the nature of the dispenser has changed?

The Nature of Statistics

The first question is easily answered once you understand something about the nature of statistical knowledge: We are using knowledge about the past to try and predict the future. When the man claims to know there's a 50% chance of shock, he is not claiming knowledge about specific future outcomes. Instead, he is claiming knowledge about the past and using that knowledge to infer the nature of the dispenser.
Here the man is making a legitimate provisional assumption: the nature of reality is consistent. In physics, we make the same kind provisional assumptions. In fact, these assumptions have names: Uniformitarianism and Methodological Naturalism. We assume that the universe has a specific nature and it will always behave according to that nature... whatever that nature might be.
One might assert that the man really doesn't "know" the dispenser has a 50% chance of shock because its not "true" anymore. This is nothing more than an appeal to the unknown to invalidate what is known. The very same  flaw we discovered in Philosophical Skepticism.
The profound truth is that all our knowledge is based on past experience. If we allow this to cripple our knowledge of the future, we are left with nothing to work with rationally. The whole point of reason, logic & thinking is to make future predictions. Without this element, "knowledge" is useless.
The only rational answer to the first question is "yes."

Something's Up

The answer to the second question is a bit fuzzier and a little misleading. There is no specific point at which the man suddenly changes his mind about the nature of the dispenser. Rather, the change becomes increasing apparent with each successive trial. A good rule of thumb statistically is that 1000 trials offers roughly a 3% margin of error... although to be fair, there are variables in this calculation that are simply not known. Weather the man averages the last 100 trials or the last 1000 trials, he will eventually arrive at the correct conclusion: the nature of the food dispenser has changed.