- #1
e12514
- 30
- 0
Is it true that
Pr( ∪_(n from m to k) ((A_n) ∩ ((A_(n+1))^c)) )
= Pr( ∪_(n from m to k) (A_n) ) - Pr( (A_k) ∩ (A_k+1) )
where A_1, A_2, ... is any sequence of sets.
Well, for the (k=m+1) case I am convinced since I can see they are equal after expanding both sides out, so for example I can see that
Pr((A∩(B^c))∪(B∩(C^c))) = Pr(A∪B) - Pr(B∩C)
but I can't manage to do the same for the (k>m) case in general, so overall I'm not convinced.
Pr( ∪_(n from m to k) ((A_n) ∩ ((A_(n+1))^c)) )
= Pr( ∪_(n from m to k) (A_n) ) - Pr( (A_k) ∩ (A_k+1) )
where A_1, A_2, ... is any sequence of sets.
Well, for the (k=m+1) case I am convinced since I can see they are equal after expanding both sides out, so for example I can see that
Pr((A∩(B^c))∪(B∩(C^c))) = Pr(A∪B) - Pr(B∩C)
but I can't manage to do the same for the (k>m) case in general, so overall I'm not convinced.