As others here have already noted, this analogy is wrong. But I think people here haven't given a convincing example of a system that behaves differently due to entanglement than it would if the behaviour were simply conditionally random (the ball example behaves identically if the balls are entangled or otherwise just classically random).
The issue in finding a good example is that the effect of entanglement is rather subtle and hard to interpret intuitively.
Here is an example that may make it more obvious:
There exists a game that can be played cooperatively between two players that share two random bits. It is possible to win this game only 75% of the time if the bits are not entangled. If the bits are entangled there is a strategy for winning the game about 85% of the time.
The details of the game and a good explanation can be found here:
https://www.scottaaronson.com/blog/?p=2464
Basically, there is a game that involves sharing two bits, if they are entangled, it can be won 85% of the time. If they are not entangled but otherwise random (like the red and blue ball example), it can be won only 75% of the time.
Here is an example that may make it more obvious:
There exists a game that can be played cooperatively between two players that share two random bits. It is possible to win this game only 75% of the time if the bits are not entangled. If the bits are entangled there is a strategy for winning the game about 85% of the time. The details of the game and a good explanation can be found here: https://www.scottaaronson.com/blog/?p=2464
Basically, there is a game that involves sharing two bits, if they are entangled, it can be won 85% of the time. If they are not entangled but otherwise random (like the red and blue ball example), it can be won only 75% of the time.