Just because Quantum Mechanics came up in this thread I thought I'd clear up a few things. There are a few interpretations of quantum theory so I'll just explain things from the perspective of the majority view of Copenhagen-style interpretations and mention some others at the end. I think QM makes more sense when compared with regular probability and betting. I've divided the post in two. The first part just deals with the basics of Probability and QM, the second part deals with entanglement.

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Firstly Quantum Mechanics uses probabilities. Probabilities are essentially a way of "managing" one's knowledge. Before somebody rolls a dice you give a probability of 1/6 to each number. If they roll the dice and you don't see the result, but you're told it was even, the probability gets updated to 1/3 for all the even numbers and 0 for all the odd numbers. The list of all the probabilities in quantum mechanics is called the **quantum state**.

This is all "state collapse" is in Quantum Mechanics. You observe something and then you update your probabilities. Quantum Mechanics also requires an observer for the same reason probability theory in general does. Not for mystical reasons, but because it makes reference to some agent and what they've learned. If I learn a fact but you don't, my probabilities change but yours do not. Just think of the dice example and somebody who doesn't learn it was an even number. So to a certain degree the theory is subjective as it takes into account the knowledge of a given agent.

One difference between Quantum Theory and regular probability is that the probabilities never go completely to 0 or 1, i.e. you never become certain of what will happen and won't happen for every experiment you could do. There is always some doubt. In regular probability it is possible in principle to eventually learn enough to remove any doubt, although it might be too difficult in practice. The fact that you will always have doubt in Quantum Theory is called the **Uncertainty Principle**. **There is a minimal level of doubt possible (our technical jargon for minimal doubt is "pure state")**

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Entanglement is a type of correlation. Nothing more.

In regular probability if two things are correlated their probabilities aren't just a combination of those for each considered on its own. There's an extra element due to their connection. However if you do enough analysis eventually you'll learn the "cause" of what is linking them and their probabilities will go back to being separate, because the only remaining doubts will be concerning each one on its own. The link has become a deterministic fact.

A very simple example would be if a computer rolls two virtual dice and adds the same common number to both of them. At first you have to model their results together, but once you learn this number you can go back to modelling them independently because the only remaining doubt concerns their individual rolls.

Entanglement is a special type of correlation between two quantum systems where basically even when you've attained minimal doubt you still haven't learned the common element that connects them and so your remaining doubt isn't just about them individually and you have to keep describing them together.

For a long time some people weren't sure if this was a problem. Whatever is connecting their results denote all the mathematical structures necessary to model it with the variable , then we just assume that the values of this are inaccessible to macroscopic study. They are "Hidden Variables". However in 1964 and more strongly in 1976 John Bell showed that such a cannot exist. What ever connects quantum systems like this is not some kind of mechanical process amenable to mathematical description.

There are some caveats to Bell's result. Three special cases his proof doesn't cover where you could recover a mathematical/mechanical understanding.

- He does assume that every experiment has a single outcome. Remove this and his proof doesn't go through. Allowing experiments to have multiple outcomes is Many Worlds

- The could violate Einstein's Relativity, returning to a world to having a Newtonian structure of absolute time.

- The could involve interaction between past, present and future in some sense.

However these are doubted by the majority. The first one has failed to match experiment since it was conceived in 1957. Every attempt to do so fails in some manner or is circular for technical reasons. The latter two are subject to a problem uncovered in this 2012 paper (

https://arxiv.org/abs/1208.4119) verified independently by others later. They require enormous fine-tuning.

So the majority opinion today (as was the conclusion of Bohr and others by the early 1930s) is that whatever is going on to connect them is not mechanical or not amenable to mathematical description. This goes hand in hand with later discoveries (e.g. Kochen-Specker theorem) that the properties QM describes (Momentum, Position, Energy, etc) seem to be properties of our experimental devices not of microscopic objects themselves.

**So QM is about how you "bet" on macroscopic impressions of the microscopic realm, but doesn't directly describe the microscopic itself which seems to be non-mechanical/non-mathematical.**