The analogy is not correct - and it cannot be, because you describe a perfectly valid classical experiment. The problem is that no analogy can be correct, because quantum entanglement is not somthing we can describe by classical physics - hence the need for quantum theory.
The first problem with your experiment is that you have fixed the measurement basis: the "color" of the apple. The second is that you only introduced classical randomness (something randomly picking an apple).
Let's focus on the first problem: If we do the same experiment with an entangled pair of spins (e.g. two particles), then the crucial observation is not that whenever Alice measures "spin up", she knows that Bob will also measure "spin down", but that regardless of which basis Alice chooses ($L_x,L_y,L_z$ or any other), the outcome of her and Bob's measurement are the same.
In a classical system, this cannot be achieved. Somehow, you'd have to be able to also say that an apple is between red and green, i.e. that it could also be of a different kind of colour (let's call it "gred" and "reen") and when Alice gets the apple, she can decide to have a look whether it's red or green, or whether it's gred or reen. Now, whatever Alice chooses to measure, the apple of Bob needs to be the same if Bob measures in the same basis. However, an apple can obviously only have ONE color at a time and whatever it was, it was well-defined from the start (if the apples in the box are green and red, they are green and red, if they are gred and reen, they are either gred or reen, but then, they are not red or green). This is different for the quantum particle.
Let's investigate your experiment in more detail: You have a box of apples with either red $apple_{red}$ or green $apple_{green}$ apples (these are only symbols). You splice the apples and send one half to Alice and one half to Bob, thereby the apples the two have are either $apple_{red,red}$ or $apple_{green,green}$. You said that the probability for having a red or a green apple is exactly one half. This means your system has the state:
$$\rho=\frac{1}{2}apple_{red,red}+\frac{1}{2}apple_{green,green}$$
Now, you measure your system, which means that Alice has a look at the first color of the state (either red or green). If she finds $apple_{red}$, she knows the apple must have been $apple_{red,red}$.
Let's have a look at the quantum system. Here, in order to see entanglement, the state of the apple must be neither red, nor green, it must be all of it. There is somehow only one type of apple and its all $apple_{red+green}$ so that after the splicing, you get a state
$$\rho=apple_{red,red+green,green}.$$
And now, Alice can decide what she wants to measure: If she decides to measure "red" or "green", she simply has a look at whether the apple is red or green - and that happens with probability 1/2. In any case, Bob will measure the same. But maybe, Alice wants to have a look at whether the state is reen or gred. Her outcome will be "reen" with some probability and "gred" with another (meaning that the apple really looks "reen" or "gred", whatever that is. It's not actually "red" or "green" and we just try to decide which of the other colors fits better - no, it's actually "reen" or actually "gred") - and once again, Bob will measure the exact same thing if he also has a look at whether his apple is "reen" or "gred". Here, you also see the "quantum probability": We don't have a box of either red or green apples with some probability each, we have a box with only one type of apple. Yet, when we have a look at the color of the appley, the outcome might be random - something that cannot happen in classical physics.
The correlations between the entangled states are therefore much stronger than in your experiment!
To summarise: To explain a maximally entangled state, you'd have to start out with a box of completely identical apples of a certain type. Then you split the apple and get one half to Alice and the other to Bob. Now Alice decides what pair of colours she wants to measure and measures it. Whatever the outcome, she then knows what the outcome will be on Bob's side.
The differences to your experiment: Alice has several choices for colors. The apple can not just have one out of two colours, but it can have a multitude of colours - however Alice can only ever measure a pair at the same time. Also, instead of a mixture of differently coloured apples, we use only one type of apple that has an undefined colour that gets only defined when Alice has a look.
Mathematically speaking: What you describe is not an entangled state, it's a maximally mixed state - and those behave very differently (see What is the difference between maximally entangled and maximally mixed states?)
