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Topic: Probabilities not in [0,1]?
Replies: 8   Last Post: Mar 12, 2013 10:56 PM

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 dan.ms.chaos@gmail.com Posts: 409 Registered: 3/1/08
Re: Probabilities not in [0,1]?
Posted: Mar 11, 2013 3:27 PM

On Mar 10, 8:36 pm, "Peter Percival" <peterxperci...@hotmail.com>
wrote:
> Is there a theory of probability in which probabilities do not lie in the
> real interval [0,1]? More specifically, is there one in which the "space"
> of probabilities may have points x, y, z with x < y, x < z but y and z not
> necessarily comparable?
>
> --
> Using Opera's revolutionary email client:http://www.opera.com/mail/

Yes for the first question , not that I know of for the second one .

More specifically , you can have negative probabilities and over-
unitary probabilities , as long as all the experimentally observable
probabilities are between 0 and 1 .

Imagine you have two socks hidden in two drawers , set up for an
experiment :
The socks can be black or white in color .

You can ask only one of three questions :
Is the sock in drawer 1 white ?
Is the sock in drawer 2 white ?
Are the socks in drawers 1 and 2 of different colors ?

And for any of those questions ,you wold get an answer 'yes' 100% of
the time . How do we model this?
We have four possible configurations corresponding to four
probabilities :
(white, white)-> p1
(black,black) -> p2
(black,white) -> p3
(white,black) -> p4

p1 + p2 + p3 + p4 = 1 (total probability)
p3 + p4 = 1 (the socks are always different)
p1 + p4 = 1 (the socks in the first drawer are always white)
p1 + p3 = 1 (the socks in the second drawer are always white)

We obtain :
(white, white)-> 1/2
(black,black) -> -1/2
(black,white) -> 1/2
(white,black) -> 1/2

This might seem absurd , but experiments to which this model is
applicable are conducted in quantum mechanics .
http://en.wikipedia.org/wiki/Bell%27s_theorem#Bell_inequalities
http://en.wikipedia.org/wiki/Negative_probability
http://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution