What you're describing is a local hidden variable theory. In this case, the "hidden variable" is who has which fruit. It's "local" because there's no remote inf

What you're describing is a local hidden variable theory. In this case, the "hidden variable" is who has which fruit. It's "local" because there's no remote influence -- Bob eating his fruit doesn't change which one Alice has. Bell's theorem places limits on the correlations that a theory like this can explain. Here's a simple example of a Bell violation: 1) Alice and Bob get together and conspire. They can do anything they want here. 2) Alice and Bob are then taken to separate interrogation chambers, to be quizzed by separate interrogators. 3) Each interrogator randomly (and independently) chooses one of three yes-or-no questions to ask their subject. Both interrogators are working from the same list of 3 questions. Alice and Bob are asked the questions and the questions and their answers are recorded. 4) This whole process is repeated many times. When the results are tabulated, it's found that every time Alice and Bob were asked the same question, they gave the same answer. But when they were asked different questions, 75% of the time they gave different answers. How could Alice and Bob have achieved this? Classically, they couldn't. In order to always give the same answer to the same question, they'd have to (either directly or indirectly) agree beforehand on which answer to give to which question. Since there are three possible questions but only two possible answers to them, any strategy they pick must give the same answer to (at least) two of the three questions. Therefore, if their interrogators pick different questions there's at least a 1/3 chance they'll pick the questions with the same answer. But with quantum entanglement, the chance of getting the same answer from both is can be only 1/4. Something weirder is going on. (BTW, for completeness, here's how the quantum results I described can be achieved: Alice and Bob are replaced by electrons, prepared in the (entangled) singlet state. The "interrogations" are measurements of their spin (up or down) on the 0°, 120°, or 240° axes, with one of the detectors set up upside-down. This is not the way it's actually done, but it's easier to explain & see the paradox this way.)