# What is the meaning of “independent events ” and how can we logically conclude independence of two events in probability?

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 .

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suppose a 3 match series between two teams A and B.suppose team A win 1st match then result of this match will not affect next match.so these type of events are called as independent events.Result of any event doesn't bother other events –  iostream007 May 13 '13 at 14:28
@iostream007 -1. A typical example of events which one can guess are quite dependent. –  Did May 13 '13 at 15:39
Philosophically, two events are independent for an observer iff gaining the information that event $A$ occurred would not increase that observers probability that $B$ occurred. Thus independence is (i) observer dependent (I might say the events are independent, you might say they're dependent; the version of me in the future with more information might agree with you that they're dependent, thereby disagreeing with his past self) and (ii) not a statement about cause and effect. –  goblin May 13 '13 at 15:42
@user18921: why don't you post your comment as an answer?. Also can you clarify this-> "not a statement about cause and effect". Thank you. –  boywholived May 13 '13 at 15:50
Your guess is a nice try but wrong. If $B$ is a null set then $A$ and $B$ are always independent, no matter what I choose for $A$. Try going through my dice example again, but this time instead of needing a six you need a four five or six. My argument works unchanged. –  Tim May 14 '13 at 12:18

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.

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Excellent!. Thank you. –  boywholived May 13 '13 at 15:36
This is a nice answer, and clarifies the distinction between our intuition about probability and the mathematics of probability. However, I don't understand the last sentence. Can you clarify why "an assumption I make to simplify my model which is usually wrong but not that wrong" is a better definition? What does "model" mean here? –  CommonerG May 13 '13 at 21:48

Philosophically, two events $A$ and $B$ are independent for an observer iff gaining the information that event $A$ occurred would not increase that observers probability that $B$ occurred. Thus independence is observer dependent - I might say the events are independent, you might say they're dependent; and the version of me in the future with more information might agree with you that they're dependent, thereby disagreeing with his past self.

Now there are events that most people will agree are dependent. The event that a coin shows a heads and the event that the coin shows a tails are almost certainly dependent; if I see a heads, my probability that the coin is showing a tails suddenly plumets to zero. But even in this example, if I already knew the coin was biased such that heads always shows, then I always knew the probability of tails would show is/was $0$, and thus my probability of seeing tails wouldn't change upon noting that heads is showing. So even here, it's possible that, for some observers, heads and tails will be independent events.

Also, at any given moment in time, there are events that most observers situated at that moment in time will regard as independent. Like if people are gambling at a craps table, then the numbers that show on the die the next time they're rolled will tend to be viewed as independent of the numbers that show on the die the time after that. Of course, this isn't entirely true. The dice may have a non-negligible bias, after all; in which case, the numbers that show after the first throw can be used to make a better prediction on the second throw. But dice tend to be pretty fair, and most sensible player's will tend to model the dice as such, and thus the numbers showing on each roll will tend to be treated as independent from one another. Note that we're thinking about idealized players here - these are players free from human flaws e.g. they certainly don't believe in the law of averages.

Finally, a word about cause and effect. If we flip two coins simultaneously, well neither can have a causal effect on the other, so we'd likely model the outcome of one flip as being independent of the outcome of the other. But, what if we knew they both came from the same factory, either factory X or factory Y, and factory X's coins always come up heads and factory Y's coins always come up tails? Well, the coins still have no causal effect on one another - but, lo and behold, the outcome of one coin allows us to predict the outcome of the other. That is, from the point of view of an observer that knows that both coins come from the same factory, either X or Y, and that those factories make systematically biased coins, the outcomes of the two coins are utterly dependent, even though there is no cause/effect relationship between them.

So, do not confound causal independence with probabilistic independence! Yes, these concepts do have some bearing on one another, but they're distinct concepts.

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Excellent answer!. –  boywholived May 14 '13 at 6:28

I think you are asking why is the notion of independent events when there is no relation between them!

Notion of independent events help to calculate the probability of occurring two independent events on like p(A)Up(B),P(A) intersection P(B) on contrary to calculating dependent events.

Independent events--> their probabilities of occurring is not dependent on one another Dependent events ---> Calculation of probabilities dependent.

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As pointed out, the question you pose is both mathematical (in which case the answer is clear and unambiguous) and philosophical (in which case, it isn't).

For a starter on the philosophy of probability - start here.

Your statement $\Large\color{red}{\star}$ is a sufficient condition is correct but it is highly restrictive. The statement $\Large\color{blue}{\star}$ is a necessarily and sufficient condition - indeed the necessarily and sufficient condition.

Consider the two events "Arsenal will win their next game" and "Manchester United will win their next game". Are these events independent? If they are playing each other - clearly not, if they are playing other teams then clearly yes.

So there exists a third event "Arsenal are playing Manchester United in their next game". The first two events are clearly dependent on this event.

The probability of this third event is interesting because it strikes at the heart of how one interprets probability.

This probability is either 0 or 1 - I could go online now and find out. But if I do not know - can I determine a probability for it that is not 0 or 1. If I can, how can this be interpreted? Further, what if nobody knows (e.g. the draw has not yet been determined)?

For some interpretations of this state of indeterminacy see Schrodinger's Cat

I think the point is that mathematics is mathematics, physics is physics, music is music and dance is dance - what they mean is down to humanity.

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