Assume we randomly pick a color ball from a bag, which is known to have a black ball and red ball. Our knowledge of the color of the selected ball will affect the selection for the color of ball in the bag immediately (faster than light?).
Does this have any relation to the entanglement, or the delayed choice phenomenon? Or is it just plain old probability?
For me, knowledge of an event could affect probability for other events, but not the other events themselves.
First of all, entanglement is a correlation property, therefore you can talk about it only in the context of several objects. You may have two bags of balls, or take several balls from the same bag.
Secondly, when you are talking about the balls, you may have a perfect correlation without any entanglement. Imagine two bags of balls, which contain the two kinds of balls you described. You may have perfect correlation between the colors of the balls you draw from this bags, i.e. when you draw a red ball from the first bag, the ball from the second will always be also red. And what I mean here is not that this is easy or even possible to arrange, but that such situation can be described by a proper joint probability distribution.
Now to add entanglement to this picture you need to introduce somehow the basis change. For a quantum object it is easy. Such an object can be in a superposition - i. e. not "red" or "black" but of an unidentified "color" between those. It only decides of which "color" it is when you perform a measurement. You may have your "quantum ball" in different states, depending on what color is more probable to get in the measurement. And the measurement itself can be done in different "basis": for example you may ask not only if the ball is "red" or if it is "black" but if it is in a particular superposition of those (and you always get a positive answer if you guess the parts correctly). The most comprehensible example of this is the light polarization. The photon can be polarized horizontally or vertically, or at any angle between those. Also you can measure if it is horizontally, vertically or say diagonally polarized and get a perfect match when the polarization angle matches your measurement angle. The change of the angle at which you measure the polarization is the measurement basis change. For a macroscopic object it is much harder if possible at all to imagine such concept.
Now if you perform the measurements of the "colors" of your balls from different bags, constantly changing the basis of the measurements on both sides, you may encounter a situation, when despite the measurement in different basis, the outcomes on both sides are correlated. Moreover such correlation can't be explained without either assuming that the balls signal to each other faster than light or introducing a new concept to explain this correlation. As faster than light communication would lead to many other paradoxical observable consequences, which by now have never been seen, the concept of entanglement is used to explain this.
It is just plain old probability. You need to read Probability theory: Logic of Science by E.T. Jaynes. Probability of any event is always conditional on the current information you possess. I will give an example from Jaynes' book, which will sound weird unless you look at it from the "conditional on information" perspective.
Suppose there are 100 balls in a bag, one of which is Red while the rest are Black. Your friend picks a ball one after another, without replacement. You are not shown what he picked. After first pick, you are asked what the probability is that that the first pick is Red. You say 1/100, which is correct. After say tenth pick, you are asked the same question again, and the correct answer is still 1/100. But suppose that you are now shown the tenth pick, and it turns out to be Red. Now the same question is asked of you again: what is the probability that first pick was Red? That probability now suddenly drops to 0. In other words, what happened on tenth pick has influenced the probability of first pick, earlier in time!
