The problem is that you missunderstand the concept of probability. Probability is, by definition, related to a single experiment. Imagine flipping a biased coin, which lands on its head side 99% of times. Bias is property of the coin. It is inseparable from the coin. Flipping the coin does nothing to change that bias. Probability of the coin landing on heads (or tails) side is quantification of that bias. Therefore it is independent of the number of experiments (tosses) performed. It is an intrinsic property of the coin. At each toss, the coin has probability p=0.99 of heads. The concepts that relate probability and the number of experiments are expected value and variance of the experiments. They are derived from probability, not vice versa.

If you think of probability in terms of outcomes and outcome space, it is quite clear. We define outcome space S consisting of disjunct outcomes o and state the following:

1. For each outcome o in S, 0 <= p(o) <= 1

2. p(S) = sum(p(o)) = 1

Thus we define probability.

We can interpret probability of an outcome o p(o) independently of the frequentist approach as a degree of belief that outcome o is true.

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I don't deny any of that. Probability is the average of a large data pool. The more we know how probable something is, the better judgement we can make. What probability cannot do is satisfactorily resolve a single event. The problem with the Monty Hall experiment is that it provides a scenario where, on one run, the probabilities are mostly irrelevant. They exist and are real, but bare no relation to a single one-off event. I'm finding it difficult to find an example of how to express myself what with having to keep coming back to answering how it relates to the complexities of the door puzzle. The door puzzle only exists because it's a repeatable pattern and because it's had the added complexity of the free goat reveal, giving the impression that you get two doors for one, when, in reality, you're just making a placeholder for a 50/50, should it be a one-off non-repeatable event.

Lets try and transfer this to RPGs, something a bit more relatable:

You're up against the end boss, at the end of the game. During your missions you've acquired an item which you can chuck at the boss. This item says on its description that it has a good chance to hit [bypassing any saves automatically].

This is a single use item that once used you have no more of them.

You have a binary result - it either hits or it doesn't.

The chance to hit might be 66%, but that is meaningless to you, either it hits or it doesn't, it is, in effect, a 50/50.

What makes it 66% is that over the course of 100 playthroughs you'll have hit the boss with it 66 times. So if you play the game 100 times you can apply a number to the item and imagine 66% instead of "good".

The graph for end results will show that on one result you got a binary, a 50/50, but as each playthrough was resolved, that the graph line will gradually smooth out into a constant of 66/33.

So the closer you are to the smaller sample size, the more the results 'appear' 50/50. And they appear that way because the 'random element' is affecting the probability. Probability seeks to put a number to random, when random is inherently chaotic and uncategorisable in smaller sample sizes.

So while probability can put a value on the singular by examination of the plural, it is completely incapable of quantifying the random nature of small sample sizes, in that, the fewer events that take place, the less relevant the probability is.

And it's a problem for mathematicians because there's no formula to measure the quantifiable element of random in any given low-sample-size scenario. This is what gives rise to paradoxes such as the door puzzle where there is a known probability, but that probability is irrelevant for a single event. Which is likely what stumped people for so long. But now that the puzzle is 'solved', the same difficulty is being experienced in trying to visualise the puzzle without the knowledge of the plural.

I really can't think of a way to describe this any better, it's such a unique kind of puzzle with very few relatable comparisons, so I'll leave it that yes, you can have your 66%, but that I disagree that on a one-shot of the puzzle it makes any difference at all.

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