Independence of a collection of events

We will say that a family of events are independent if knowledge about some of the events doesn't change my beliefs, my probability model, for the remaining events.

A collection of events are independent if you can calculate probabilities of intersections of these events by multiplying individual probabilities. And this should be possible for all choices of indices involved and for any number or events involved.

If A1，A2, A3 are independent. The probability of event A3 is the same as the probability of event A3, given that A1 and A2 occurred. Or the probability of A3, given that A1 occurred but A2 didn't. Or we can continue this similarly, the probability of A3 given that A1 did not occur, and A2 occurred, and so on. So any kind of information that I might give you about events A1 and A2-- which one of them occurred and which one didn't-- is not going to affect my beliefs about the event A3. The conditional probabilities are going to be the same as the unconditional probabilities.