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. (01)
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. (02)
(Ignore if I am getting this wrong.) (03)
vQ (04)
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