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| OsherD... |
 Posted: Thu Nov 19, 2009 3:34 pm |
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From Osher Doctorow
We now have established that:
1) If A, B are disjoint (do not intersect), then P ' (A --> B) - P(A--
[quote]B) = P(B)
2) If A, B intersect in a set AB of probability > 0, then P ' (A-->B)[/quote]
- P(A-->B) = P(A ' B)
3) If B is a subset of A (with probability 1), then P ' (A-->B) - P(A--
[quote]B) = 0.
[/quote]
These follow from:
4) P ' (A-->B) - P(A-->B) = [1 + P(B) - P(A)] - [1 + P(AB) - P(A)] = P
(BA ' ) = P(A ' B)
since in case (1) B is a subset of A ' so P(A ' B) = P(B), and in case
(3) B is a subset of A so P(A ' B) = 0.
We rewrite (1), (2), (3) respectively as:
5) P ' (A-->B) = P(A-->B) + P(B)
6) P ' (A-->B) = P(A-->B) + P(A ' B)
7) P ' (A-->B) = P(A-->B)
Since (6) holds for all 3 cases (the other 2 are special cases of (6),
P ' (A-->B) adds to P(A-->B) a quantity that increases with the
separation of A and B, so:
8) P ' (A-->B) - P(A-->B) = Separate PI = "SPI"
Notice that the more A and B intersect, that is to say overlap, the
more P(A ' B) approaches 0 from the right, while the less A and B
intersect, the more P(A ' B) approaches P(B) since more of B becomes a
subset of A ' .
Finally, P ' (A-->B) can be interpreted as follows:
9) P ' (A-->B) is P(A-->B) + SPI = PI + SPI where P(A-->B) is taken
as PI.
So the second type of PI, namely P ' (A-->B), is the first type, P(A--
[quote]B), plus a "Separation PI" which is SPI.
[/quote]
Readers can begin to appreciate the physics applications when we
recall the elementary particle description of Interactions as "fermion
A transmits boson B to fermion C", etc., or in wave-particle or field-
particle analog, A moves closer to C, intersects C in B, and then
separates, although the letters have to be modified somewhat to
correctly describe this analogy.
Osher Doctorow |
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| OsherD... |
| Posted: Thu Nov 19, 2009 3:40 pm |
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From Osher Doctorow
For those who skimmed the previous post too rapidly, note that:
1) SPI = P(A ' B)
Osher Doctorow |
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