random

[David C. Ullrich]

On Wed, 6 Jun 2001 18:35:03 +0200, "Alex Martelli" <aleaxit at yahoo.com>
wrote:

>"David C. Ullrich" <ullrich at math.okstate.edu> wrote in message
>news:3b1e3e18.160819 at nntp.sprynet.com...
>    ...
>> Answer the question instead of only replying
>> to the parts you feel like replying to.
>
>This thread is long enough without going around
>commanding each other what to answer, but for
>once I'll make an exception:-) [not fully
>responding to this post -- yet -- but only to
>the 'mandatory' parts:-)]:
>
>> _Do_ you think it makes sense to talk about
>> the probability that the first digit of pi
>> is 3?
>
>Yes! 

Fabulous. Let's assume for the sake of argument
that pi is normal - it may well be, and it's 
a number that we have a name for.

Pi "is" normal. If this says something about
_probability_ of digits of Pi being this or
that then the probability of the first digit
being 3 is 1/10. (Note that by "first digit
I mean the one before the decimal point.)

Have you got all the money you need? I have
a deal for you: I'm willing to _bet_ that
the first digit of Pi is 3, for just about
any stakes you name (with suitable escrow
arrangements if we start talking about large
sums.)

Since the probability that the first digit
is 3 is just 1/10 I should really ask for
9 to 1 odds here - that would make it a fair
bet. But silly me, I'm willing to offer
you even money on this! This is an offer you
can't refuse, getting even money on a bet
where you actually have a 9 to 1 advantage.

> Or rather it makes sense to talk of
>    P(first digit of pi is 3 | y)
>for any y representing (if I recall De
>Finetti's formula correctly) "a self-consistent
>set of beliefs about the state of the Universe".

Um. Can you describe for me a self-consisent
set of beliefs about the universe y with the
property that P(first digit of pi is 3 | y)
is something other than 1?

Never mind. Don't tell me what self-consistent
set of beliefs about the universe makes
P(first digit of pi is 3 | y) = 1/10.
But please do adopt such a set of beliefs.
I could use the money...

>I do not believe that it REALLY makes sense
>to talk about any P(x), rather than P(x|y),
>unless the y can reasonably be taken as
>given by the context.  I do realize that for
>some strange reasons this De Finetti idea
>is not universally accepted (yet:-), but
>it IS widely accepted enough among scholars
>of probability theory that I don't think I
>need to defend it in depth -- or is my hope
>unfounded and do we need to branch off into
>ANOTHER huge subthread about whether
>_unconditional_ probabilities "exist"...?-)

Heh-heh. If it's hard or impossible to
pin something down then we'd better not
talk about it, and hence the things you've
said about probability must be right.
(No, you didn't quite say that.)

>> (Or rather: Does it really make sense
>> to say that the probability that the first
>> digit of pi is 1/10?)
>
>It surely "makes sense" (like many false
>statements can "make sense") but may be
>false for any y, if we carefully constrain
>all relevant definitions (i.e., it is quite
>possible that no y such that the P(x|y),
>for the x we've been talking about, which
>is self-consistent in the De Finetti sense).

Assuming still that Pi _is_ normal: You've
_said_ that a normal number _does_ give
an RNG satisfying the definition I gave,
that is [c] independent digits, each with
probability 1/10 (that's the decimal version).
I'm not sure _exactly_ what the previous
paragraph means, but it seems like you're
retracting that - you no longer insist that
it _is_ _true_ that the probability that
the first digit of Pi is 3 is 1/10?

>Alex
>
>
>



David C. Ullrich
*********************
"Sometimes you can have access violations all the 
time and the program still works." (Michael Caracena, 
comp.lang.pascal.delphi.misc 5/1/01)