Intersection of two independent events

1. Aug 14, 2013

Avichal

If A and B are two independent events then P(A intersection B) = P(A).P(B)
I don't refute this but it confuses me. What is the sample space in this?
For eg: - If A is the event that we get Head while tossing a coin and B is the event that we get 2 while throwing a die, then what will we be the sample space? Is it {H, T} or {1, 2, 3, 4, 5, 6}?

2. Aug 14, 2013

chiro

Hey Avichal.

If you have two sets A and B that are independent, then the state space for all combinations is the Cartesian product C = A X B (a belonging to A, B belonging to b, c = (a,b) belonging to C).

With regards to independence, the way you get the definition is the following:

P(A|B) = P(A and B)/P(B) [definition of conditional probability].

If A is independent from any other random variable then P(A|B) = P(A). This implies:

P(A|B) = P(A) = P(A and B)/P(B). Multiplying everything by P(B) gives

P(A and B) = P(A)*P(B) and thus proved.

3. Aug 14, 2013

Avichal

Hmm ... If A and B are dependent then P(A|B) = P(A and B) / P(B)
Here what will be the sample space? For B it will be different compared to A and B. So how does it work?

4. Aug 14, 2013

mathman

The sample space is the direct product space, i.e. the elements are {h,1}, {t,1}, {h,2}, etc.

5. Aug 24, 2013

haruspex

Hi Chiro,
I think that's a bit confusing. Perhaps you meant this:
If you have two sets A and B that are subsets of apparently unrelated event spaces Ω1, Ω2, then in order to discuss joint probabilities etc. you must first combine the event spaces. Given their independence as spaces (not to be confused with independence of events within a space), the appropriate combination is the Cartesian product ΩC = Ω1 X Ω2. Each space has its own probability function, but they are related. A ⊂ Ω1 maps naturally to A X Ω2 ⊂ ΩC. P1(A) = PC(A X Ω2) etc. Now we can understand the joint event "A and B" as (A X Ω2) ∩ (Ω1 X B) = A X B.

6. Aug 24, 2013

chiro

I was only talking about the event space without reference to a probability measure. The measure and filtration/sigma-algebras come on top of this, but I wasn't really getting specifically into this (as you have).

This was just a way to say that if you can have every permutation of possibilities from both sets, then the cartesian product will give you a state-space that describes such possibilities.

Applying zero probabilities or other constraints is further step on top of this.