> As I have heard, entanglement doesn't provide FTL comms. [...]

Right, if by communication we mean some system where one party can specify a bit value and have that bit value be transmitted to some other party.

> But it's not magic, it's just how probabilities work as we learn more information about a system.

No, it is much more than that. A good illustration of entanglement occurs in the CHSH game.

Here's how the game is played. You and I are playing as a team against the house. We are taken to separate locations, very far apart. There is a game master at each of the locations. We each play 1000 rounds of the game with the game master at our location.

Each round consists of the game master stating the round number, and then flipping a (completely fair and perfectly random) coin and revealing the result (H or T). The player then says "1" or "0". The game master writes down the round number, the coin flip result, and the number the player stated.

After 1000 rounds, we all return to a common location, and the score is calculated. For each round, we get a point if either (1) both game master's coins came up H and we picked different numbers, or (2) at least one game master's coin came up T and we picked the same number. The higher our total score for the 1000 rounds, the bigger our reward.

Before we are taken to the separate locations and play starts, we are given as long as we want to plan how we want to play. We can make any preparations we want, and bring anything we want with us. The only constraints are that we are not allowed to do anything that will mess with the coin flips. We will be far enough apart that the speed of light limit stops any communication between us during play.

If we adopt the simple plan of "always say 1", we'll score a point in 75% of the rounds. Another simple plan is that I always say 0, and you say 0 on T, 1 on H. That also scores 75% of the time for us. Can we do better?

With a little thought, you can probably convince yourself that we cannot. In a world without entanglement, that would be correct.

With entanglement, we can score in 85% of the rounds!

Consider a photon that has just come through a polarizing filter set at a 0 degree angle. That photon is polarized at 0 degrees. If you try to send it through another polarizing filter also set at 0, it goes through. If the other filter is at 90 degrees, the photon is blocked. If the other filter is at some angle in between, say T, then the photon goes through with probability cos(T)^2, and if it does go through, it is now polarized at T.

What we do is prepare 1000 pairs of photons. The two photons in each pair are polarized the same way and entangled. We number these pairs from 1 to 1000, and you take one from each pair and I take one from each pair.

Now when we play the game here is what we do. When your game master flips his coin and shows you the result, you take your photon for that round and send it through a polarizing filter. You set the filter to 0 degrees if the coin came up H, and 45 degrees if the coin came up T. If the photon passes through the filter, you say 1, else say 0.

I do almost the same thing. The difference is I set my filter at 22.5 degrees if my game master's coin is T, and 67.5 degrees if it is H.

Let's look at what happens. In the following I'll assume you send your photon through your filter before I send mine through my filter, but it doesn't actually matter who goes first (or even if we happen to act simultaneously). It is just easier to talk about if we do it sequentially.

Suppose you see H and I see H. You measure with the filter set at 0. If your photon gets through (and so you say "1"), its polarization becomes 0, and since mine is entangled with it mine also becomes 0. When I measure with my 67.5 degree filter, there is only a cos(67.5)^2 chance (15%) my photon also gets through, and a sin(67.5)^2 chance (85%) mine gets blocked. So, 85% of the time you say "1" I say "0" in the H/H case. Remember, we want to say different numbers on H/H, so this is good for us.

Same on H/H if your photon gets blocked and you say "0". Because they are entangled, my photon becomes polarized at 90 degrees (so that it would also be blocked by a 0 degree filter), and when I measure it with a 67.5 degree filter, the difference between my filter angle an the photon is 22.5 degrees, so it will pass my filter cos(22.5)^2 of the time, or 85%.

Here's a little table to help see what is going on here:

You Me
H 0
22.5 T
T 45
67.5 H

The "you" column shows what angle you set your filter to for each coin outcome. Second column is for me. They key here is that when we both see H, we are setting our filters 67.5 degrees apart, and so the probability that they will produce the same outcome is cos(67.5)^2, which is 15%, and so we win 85% in the H/H case (remember, we want to mismatch in that case). When one of us sees a T, our measurement angles differ by 22.5 degrees, so we match 85% of the time.

If we were NOT using entangled photons, this would not work. Suppose, for instance, that all the photons were polarized at 0 degrees and they were not entangled. In the H/H case and H/T case, we'd still do well (85%). On T/H and T/T we'd bomb. You would be measuring a 0 degree photon with a 45 degree filter, and it is 50/50 whether it goes through or not. You are effectively just flipping a coin, and nothing I do matters--we win half these and lose half these. Our overall win rate is only 67.5%, which is worse than if we had went with "always say 1" and not bothered with all this photon crap.

Only with entangled photons are we able to beat 75%. If I see heads and so measure at 67.5 degrees, it was something that happened when you measured that "told" my photon whether it should have a high or a low probability of making it through my filter. Something happened FTL after you made your measurement that let my photon "know" whether or not it should go through the 67.5 degree filter with an 85% chance or a 15% chance.

This cannot be explained with models like your red and blue marble example, where all the state is finalized when the marbles are together and then it is simply revealed to us when the marbles are far apart. In the CHSH game, the state is not finalized until after the photons are far apart, because it depends on our actions after we see the coin flips.