uniform distribution (discrete and real)

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November 26th 2010, 02:03 AM
Mobius

uniform distribution (discrete and real)

Two questions about uniformly distributed variables:

1. Let X be a random real number between 0 and 1. What is the probablity of X's decimal notation containing 123 anywhere? Intuitively I feel that this probability should be 0, because I think there are infinitely more numbers in (0,1) without 123 in their decimal notation.

2. Let X be a random natural number. What is the probability of X being even? Intuitively I feel that this probability should be 1/2, because there are just as many odd as even natural numbers.

I'm having a hard time proving either 'officially'..?
November 27th 2010, 01:13 PM
Focus

Quote:

Originally Posted by Mobius

Two questions about uniformly distributed variables:

1. Let X be a random real number between 0 and 1. What is the probablity of X's decimal notation containing 123 anywhere? Intuitively I feel that this probability should be 0, because I think there are infinitely more numbers in (0,1) without 123 in their decimal notation.

2. Let X be a random natural number. What is the probability of X being even? Intuitively I feel that this probability should be 1/2, because there are just as many odd as even natural numbers.

I'm having a hard time proving either 'officially'..?

First of all number 2 is nonsensical because you cannot have a uniform distribution on the natural numbers.

The answer for 1 is non zero. It is the size of intervals of the form $[123\cdot10^{-n},124\cdot10^{-n})$ for n>2. You may end up counting some of them more than once like this but I think those should be at most countable.
November 27th 2010, 07:58 PM
CaptainBlack

Quote:

Originally Posted by Focus

The answer for 1 is non zero. It is the size of intervals of the form $[123\cdot10^{-n},124\cdot10^{-n})$ for n>2. You may end up counting some of them more than once like this but I think those should be at most countable.

If fact the answer to the first is 1, since almost all reals are normal almost all contain 123 in their decimal expansions infinitely often.

CB
November 28th 2010, 03:58 PM
Mobius

Quote:

Originally Posted by CaptainBlack

If fact the answer to the first is 1, since almost all reals are normal almost all contain 123 in their decimal expansions infinitely often.

This surprises me. My reasoning was this: for every real number that contains 123 in its decimal notation, there are corresponding real numbers with their occurances of 123 replaced by 1243, 12443, 124443, 1244443, etc.

So for every real number containing 123, there are infinite more numbers not containing 123 (of which the example with the additional 4's are just an infinitesimal small fraction).

Where am I thinking wrong?
November 29th 2010, 03:57 PM
Focus

Quote:

Originally Posted by Mobius

This surprises me. My reasoning was this: for every real number that contains 123 in its decimal notation, there are corresponding real numbers with their occurances of 123 replaced by 1243, 12443, 124443, 1244443, etc.

So for every real number containing 123, there are infinite more numbers not containing 123 (of which the example with the additional 4's are just an infinitesimal small fraction).

Where am I thinking wrong?

The size (or cardinality) of the set doesn't really matter. You can have uncountable sets that have measure zero.