[Python-ideas] random.choice on non-sequence

[David Mertz]

For any positive integer you select (including those with more digits than
there are particles in the universe), ALMOST ALL integers are larger than
your selection. I.e. the measure of those smaller remains zero.
On Apr 13, 2016 6:31 PM, "Steven D'Aprano" <steve at pearwood.info> wrote:

> On Thu, Apr 14, 2016 at 09:47:37AM +1200, Greg Ewing wrote:
> > Chris Angelico wrote:
> > >On Wed, Apr 13, 2016 at 8:36 PM, Terry Reedy <tjreedy at udel.edu> wrote:
> > >
> > >>On 4/13/2016 12:52 AM, Chris Barker - NOAA Federal wrote:
> > >>
> > >>>BTW, isn't it impossible to randomly select from an infinite iterable
> > >>>anyway?
> > >>
> > >>With equal probability, yes, impossible.
> [...]
>
> > I think Terry meant that you can't pick just one item that's
> > equally likely to be any of the infinitely many items returned
> > by the iterator.
>
> Correct. That's equivalent to chosing a positive integer with uniform
> probability distribution and no upper bound.
>
>
> > You can prove that by considering that the probability of
> > a given item being returned would have to be 1/infinity,
> > which is zero -- so you can't return anything!
>
> That's not how probability works :-)
>
> Consider a dart which is thrown at a dartboard. The probability of it
> landing on any specific point is zero, since the area of a single point
> is zero. Nevertheless, the dart does hit somewhere!
>
> A formal and precise treatment would have to involve calculus and limits
> as the probability approaches zero, rather than a flat out "the
> probability is zero, therefore it's impossible".
>
> Slightly less formally, we can say (only horrifying mathematicians a
> little bit) that the probability of any specific number is an
> infinitesimal number.
>
> https://en.wikipedia.org/wiki/Infinitesimal
>
>
> While it is *mathematically* meaningful to talk about selecting a random
> positive integer uniformly, its hard to do much more than that. The mean
> (average) is undefined[1]. A typical value chosen would have a vast
> number of digits, far larger than anything that could be stored in
> computer memory. Indeed Almost All[2] of the values we generate would be
> so large that we have no notation for writing it down (and not enough
> space in the universe to write it even if we did). So it is impossible
> in practice to select a random integer with uniform distribution and no
> upper bound.
>
> Non-uniform distributions, though, are easy :-)
>
>
>
>
>
> [1] While weird, this is not all that weird. For example, selecting
> numbers from a Cauchy distribution also has an undefined mean. What this
> means in practice is that the *sample mean* will not converge as you
> take more and more samples: the more samples you take, the more wildly
> the average will jump all over the place.
>
> https://en.wikipedia.org/wiki/Cauchy_distribution#Estimation_of_parameters
>
> [2] https://en.wikipedia.org/wiki/Almost_all
>
>
>
> --
> Steve
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