Intersection of two independent events

[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}?

 

[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.

 

[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?

 

[mathman]

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

 

[haruspex]

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).

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 Ω_C = Ω₁ X Ω₂. Each space has its own probability function, but they are related. A ⊂ Ω₁ maps naturally to A X Ω₂ ⊂ Ω_C. P₁(A) = P_C(A X Ω₂) etc. Now we can understand the joint event "A and B" as (A X Ω₂) ∩ (Ω₁ X B) = A X B.

 

[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.