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## Main Question or Discussion Point

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.