Waclaw Kusnierczyk wrote:
> Kathryn Blackmond Laskey wrote:
>> To do probability right, you need real numbers, and you need to be
>> able to do continuous distributions. For example, consider a number
>> picked at random between zero and one. (Imagine a spinner that can
>> land anywhere on the disk with equal likelihood; measure the distance
>> around the edge to where it landed and divide by the circumference --
>> the number you get is uniformly distributed between zero and one.)
>> The probability of getting a number in any sub-interval of the unit
>> interval is equal to the length of the sub-interval. But the
>> probability of getting any specific number is zero.
>>
>> This raises some tricky mathematical issues. In finite domains, we
>> can equate probability zero with unsatisfiability and probability 1
>> with validity. But in the example I gave above, UniformRand = X is
>> satisfiable for any X between zero and one, yet (Prob (UniformRand =
>> X)) is zero for all X, and moreover, Prob(Exists X UniformRand=X))
>> is equal to 1.
>
> Isn't the issue due to approximation? In P(X = x) = 0 for any x in the
> selected interval of real numbers, what is the zero? The probability
> that a random variable will take on a specific value from a particular
> set of values is per definitionem equal to the inverse of the
> cardinality of the set -- right? An interval -- a dense set of real
> numbers, e.g., the unit interval (0.0, 1.0) -- has the cardinality
> aleph_0 (I guess). The zero in P(X = x) = 0 is in fact the inverse of
> aleph_0; it is not exactly the usual 0, it is an approximation. For
> every x in the interval, P(X = x) is infinitesimal, but is not exactly
> 0. If infinitesimals were equal to 0, would integration make sense?
> It may be practically zero, but when discussing paradoxes of
> probability, it may be worth mentioning. (01)
... thus in any non-empty interval there are aleph_0 real numbers x for
each of which P(X = x) is non-zero, and thus obviously P(exists x: X =
x) is 1. (02)
> (Ignore if I am getting this wrong.) (03)
Still holds. (04)
vQ (05)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (06)
|