The Fraction That Is Random

#1
The number of random strings of uniform distribution out of all of the possible strings of bits that are n bits in length is a fraction that would seem to depend upon n. What tools would be used to answer this question?

There are 2 raised to the n possible strings of n bits (some call it 2**n). The number of possible strings versus the number of bits in each is (2**n)/n. This ratio gets widely larger as n increases. What happens to the fraction of them that could pass the most thorough test of randomness for a uniform distribution?

I'm sorry to say, I don't know how to start.

Thank you for your help.

Jim Adrian
 
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Dason

Ambassador to the humans
#2
There really isn't a test for randomness for a uniform distribution if you only have a single observation. All outcomes are equally likely so what would that test look like?
 
#3
There really isn't a test for randomness for a uniform distribution if you only have a single observation. All outcomes are equally likely so what would that test look like?
I am thinking that the test of randomness on each possible string of n bits would be a set of auto-correlations which correlate all of the pairs at a distance of 1, then the pairs at a distance of 2, . . .all the triplets at all combinations of two distances . . . etc.

How you determine that a given string of n bits is random is not my question. If some of the possible strings of n bits are random and all others are not, then what fraction of all possible strings of n bits is random?

Thank you for your help.

Jim Adrian
 
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#4
The number of random strings of uniform distribution out of all of the possible strings of bits that are n bits in length is a fraction that would seem to depend upon n. What tools would be used to answer this question?

There are 2 raised to the n possible strings of n bits (some call it 2**n). The number of possible strings versus the number of bits in each is (2**n)/n. This ratio gets widely larger as n increases. What happens to the fraction of them that could pass the most thorough test of randomness for a uniform distribution?

I'm sorry to say, I don't know how to start.

Thank you for your help.

Jim Adrian

Is there a better forum for this question in Talk Stats?

Jim Adrian