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Re: [Phys-L] probability question

On 12/17/18 11:04 AM, Philip Keller via Phys-l wrote:

What is the probability that the (10^10^10^1000)th digit of pi is even?

This is an interesting question IMHO. It is directly relevant
to physics teaching, because it is intimately related to roundoff
error. See appendix below.

Phil also mentioned:

(For an indication of where I am leaning, I asked myself how I would answer
the question if it were about the 200th digit....)

That's an excellent starting point.

We can formalize that idea using the following maxim:
When somebody asks what's the probability,
you should ask /what's your ensemble/?

For example: What's the chance of rolling a "6" ...
... with one die?
... with two dice?
... with six dice?

Or: What's the chance that the ace of spades will be found on
the bottom of the deck ...
... when the deck comes fresh out of the manufacturer's package?
... after the deck has been shuffled?
... after the deck has been artfully sorted by some smart-aleck?

In more detail: There are a great many probability distributions
aka probability measures. Anything that conforms to a handful of
very simple axioms qualifies.

1) So, what's the chance that such-and-such digit of π is even ...
a) if you ask exactly the same question each time?
b) if you have an ensemble of similar questions?

In case (b) the answer is P=0.5.

In case (a) the answer is either 0 or 1.

2) If you don't know, then the interesting follow-up question is
how likely you are to guess correctly -- but that is a different
question! The principle here is the same: the ensemble of guesses
is different from the ensemble of actual careful calculations.


I can predict with high confidence that may people will confuse
questions (1a), (1b), and (2). This makes the topic difficult
to discuss.



Appendix: Physics connection:

1) rounding π to two decimal places is not probabilistic.
The roundoff error is the same very time.

2a) rounding noisy observations (laboratory data) is
probabilistic if the rounding isn't too coarse.

2b) coarse rounding can easily become non-probabilistic,
whereupon (2b) looks a lot like (1).

This gives us two very different answers to the question
Phil asked once upon a time: "What is uncertainty, anyway?"

This is a very big deal, because the concepts and the mathematical
tools what work in case (2a) don't work in case (1).