I would like some clarification as to if independence as referred to in both of the following cases, is the same or different. I would assume they are different?
Scenario 1: For any SINGLE trial of a random experiment, when P(A|B)=P(A) and P(B)=P(B|A). If when it is given that one happens, it doesn't change probability of the other happening, it is called independent. This seems to be the most common meaning of independence.
So let's say you had a venn diagram where P(A)=0.4, P(A∩B)=0.1 and P(B)=0.25. The probability of A is 0.4, and the probability of A given B is 0.1/0.25=0.4, so P(A)=P(A|B). And same for P(B) and P(B|A), so the events are independent.
However, I also see it used in a second scenario and I was wondering if it's technically referring to the same thing or not.
Scenario 2: When there are MULTIPLE trials/outcomes (let's say two). If the probabilities of the first trial has no bearing on the probabilities of the second, then I also see that it's called independent in certain videos and textbooks. For example, if you have 3 red and 4 green balls in a bag, and you draw one out at random, but before you take a second ball out, you put the ball back in. This is often called independent.
If you didn't place the ball back in, then you would be taking from a smaller sample the second time, and so the events would be dependent. So if you picked red, the probability of getting green afterwards would be 4/6, not 4/7.
This use of the word 'independence' and 'dependent' seems different to the one in the first scenario.
Question: Am I right in saying that the term 'independent' is technically referring to two different ideas in the two scenarios? If it's referring to the same thing, I would be confused.
(this next paragraph just explains why I'm confused about this so you can probably ignore it if you want) This question arose because a high school textbook that I'm going through seems to sort of morph these two things into one idea and it confuses me. When it gets us to use the product rule formula, P(A∩B)=P(A)P(B) for when the events are independent (single event independence), it seems to imply not only can you use this to find probability of the intersection between A and B (for one trial, meaning they happen at the same time), but also the probability of A occuring, then B occuring (for two trials, so one happens after the other), so in scenario 2, this would equate to multiplying 4/7 by 4/6 to find probability of Red then Green. Now, I can see the method is the same where you multiply the probabilities, but I'm curious as to whether this is happenstance or not. Because if they meant the same thing, that would mean that you can interpret P(A∩B) like A and B happening at the same time, but also as A happening, then B happening afterwards (which seem like two completely different things?). The textbook seems to conflate the two though.
I think Scenario 2 is a special case of Scenario 1. Choosing two balls one after the other can be considered a single random experiment. Then you have the event A ="The first ball is green" and B="The second ball is red". These are two events that can occur in this single experiment. They are independent in the sense of Scenario 1.
A more interesting fact is that whatever the colors, the events are independent. Then we start talking about independence of random variables rather than independence of events. These are two related but distinct kinds of independence.