Ytukyg

Suppose a sample space 'S'. Suppose 'E1' and 'E2' be two event sets on 'S'. Probability of 'E1' is the sum of probabilities of all sample points which are present in 'E1' (similar for E2). If E1 and E2 are independent, then the intersection of E1 and E2 be a null set isn't it ?

I cannot understand, why we are we multiplying to find intersection probability, also since the intersection is a null set , probability should be zero ? I cannot understand where my reasoning is going wrong ?

Independence does not mean the intersection is a null set. It means the chance of one event is not influenced by whether the other happens or not. Let us throw a coin and a six sided die. E1 is the event that the coin comes up heads. E2 is the event the die shows a $6$. The two events are independent because the chance of heads is the same whatever the die shows and the die does not change probabilities based on the coin. E1 has probability $frac 12$, E2 has probability $frac 16$ and the chance of both is $frac 12 cdot frac 16=frac 1{12}$

If $E_1$ and $E_2$ are independent, then the intersection of $E_1$ and $E_2$ be a null set isn't it?

NO, it isn't. You're confusing two different concepts: independent events and mutually exclusive events. (If it's any consolation, this is an extremely common mistake among students, although I don't really understand why…)

• If the intersection of two events is the empty set, then this is precisely the definition of these two events being mutually exclusive. Informally speaking: as the name suggests, either one of these events excludes the possibility of the other event happening.




• But it's a completely different concept for two events to be independent. Even before we state any definition of independence, it's very important to remember that it's not the same as being mutually exclusive, i.e. it's not at all about the empty intersection, but rather it's something else and unrelated.

Now, what does it mean for two events $A$ and $B$ to be independent? As a formula, the definition says that two events $A$ and $B$ are independent if and only if $P(Acap B)=P(A)cdot P(B)$. I'm not going to redo the whole discussion that can be found in any standard textbook and in a lot of places online (such as Wikipedia). But informally speaking, independence means that the knowledge about one of them happening or not gives us no additional information about the chances of the other one happening.

Note that, in particular, this implies that two mutually exclusive events can't be independent. If we know that $A$ and $B$ are mutually exclusive, i.e. that $Acap B=varnothing$, and that event $A$ has happened, then event $B$ definitely cannot happen. (Strictly speaking, we also need to assume that both $P(A),P(B)neq0$ for this reasoning to work.) So knowing something about one of them gives us a lot of information about the other one — which is the exact opposite of being independent!