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From: Osher Doctorow on 2 Jul 2010 00:33
From Osher Doctorow

The Operators:

1) 1ERASE(1, u, v) = (u, v) or (0, u, v) (depending on convention)
2) 1ADD(0, u, v) = (1, u, v)

Let us begin with:

3) P ' (A-->B) =1 + P(B) - P(A), P(B) < = P(A), also written P ' (A--
>B) = (1, P(B), P(A))

Then notice that:

4) P ' (A ' --> B) = 1 + P(B) - [1 - P(A)] = P(B) + P(A) = 1ERASE(1,
P(B), SGNCHG(P(A)))
= (0, P(B), -P(A)) where SGNCHG(P(A)) = -P(A).

5) P ' (A --> B ' ) = 1 + P(B ' ) - P(A) = 1 + (1 - P(B)) - P(A) = (1
- P(A)) + (1 - P(B)) = 1ADD(1, SGNCH(P(B)), P(A)) = (2, -P(B), P(A)).

When we use A, B as bounded sets/events and A ' , B ' as unbounded
complements of A, B in an unbounded space, as earlier in this thread,
then readers can see that (3) involves one bounded and one unbounded
set (1 - P(A) = P(A ' ) is unbounded, while P(B) is bounded), (4)
involves two bounded sets (P(A) and P(B)), (5) involves two unbounded
sets (1 - P(A) = P(A ' ) and 1 - P(B) = P(B ' ).

Osher Doctorow


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