What is the meaning of "independent events " in probability
For eg: Two events (say A and B)are independent , what I understand is the occurrence of A is not affected by occurrence of B .But I am not comfortable with this understanding , thinking this way every events I meet upon are independent!, does there exist a more mathematical definition( I don't want the formula associated with it)
Another thing I want to know in a real case or a problem how do we
understand(logically) if two events are independent?
That is without verifying that $P(A\cap B) = P(A)P(B) \Large\color{blue}{\star}$ how do we conclude that two events $A$ and $B$ are independent or dependent?.
Take for eg an example by "Tim"
Suppose you throw a die. The probability you throw a six(Event $A$) is $\frac16$ and the
probability you throw an even number(Event $B$) is $\frac12$.
And event $C$ such that $A$ and $B$ both happen would mean $A$ should happen(As here $A$ is a subset of $B$) hence it's too $\frac16$
My thought:
The above example suggest something like this "Suppose I define two events $A$ and $B$ and if one of the even is a subset of the other then the events are not independent". Of course such an argument is not a sufficient condition for dependency , for eg: consider an event $A$ throwing a dice and getting an odd number and event $B$ getting 6. They aren't independent ($\Large\color{blue}{\star}$ isn't satisfied) . So again I improve my suggestion "Suppose I define two events $A$ and $B$ and if one of them is a subset of the other or their intersection is a null-set then the events are not independent $\Large\color{red}{\star}$"
So at last is $\Large\color{red}{\star}$ a sufficient condition?. Or does there exist a suffcient condition (other than satisfying the formulas $\Large\color{blue}{\star}$? And what is the proof, I can't prove my statement as the idea for me is not that much mathematical .
Best Answer
The mathematical definition is very easy. Two events $A$ and $B$ are independent if and only if $$P(A\cap B) = P(A)P(B).$$
In "pure" probability theory there's no interpretation of this, it's just a definition. It's a purely mathematical statement I can make about two events and a probability distribution.
To explain what it "means" you have to explain what probability means. There's no acceptable answer to this question. It's a big philosophical problem that mathematicians avoid by writing down some equations and solving them.
The motivation comes from the idea of conditional probability.
Suppose you throw a die. The probability you throw a six is $\frac 16$ and the probability you throw an even number is $\frac 12$. You can check with the formula above that the two events are not independent.
To get an idea of why suppose you throw a die but don't look at it. You want to get a six. I tell you whether or not it's even and you decide whether to keep it or roll it again. If I tell you it's odd then you know it's not a six and you roll it again. If I tell you it's even then there are only three numbers it could be and one of them is a six. So the probability that you got a six is now one in three. So you'd be crazy to throw again.
In maths we define conditional probabilities as follows$$P(A|B) = \frac{P(A\cap B)}{P(B)}.$$ Again in "pure" maths there's no interpretation of this, it's just a formula. But in the real world $P(A|B)$ is the probability that $A$ happens if you already know that $B$ happened.
So the interpretation of independence is that $A$ and $B$ are independent if and only if $P(A|B) = P(A)$ if you know that $B$ happened it doesn't affect the probability that $A$ happened.
This concept makes intuitive sense to people. If my team is winning at half time it's more likely to win the game than if it wasn't, so not independent. If my team is winning at half time it doesn't make it less likely that it's going to rain tomorrow, so independent.
It's worth noting though that independence is an assumption that I might be wrong about. If my team happens to play well in the rain and they're winning at half time it's more likely to be raining during the game. This might mean it's more likely to be raining tomorrow. So they might not be independent.
So in fact a better definition of independence would be an assumption I make to simplify my model which is usually wrong, but hopefully not that wrong.