The randomness of randomness

The randomness of randomness

GOD does not play dice .... Albert Einstein.

It all started with a rather innocent looking question on the Internet. Here is how it ran: Given a number (only one), we can answer certain questions about it e.g. is it a prime ? is it an even number ? etc. Now, given a number (only one) can we say if it is a random number ? Why not ?

I posted the above question on the net, and got some replies, all of which gave rise to more questions.

Some of the responders were statisticians in the traditional sense. And predictably, they took examples of their pet random process: tossing a coin, or casting a die. Yes, given no other information, such experiments may be considered to give random outcomes (will it be a head ? will be a tail ? will the die yield an even number ? will the next child be a boy ? ...). But, there is a catch somewhere deep within this premise. A physicist friend of mine argued that there could be no such thing as a random number generated by such devices (die, coin). After all, the outcome of a toss is bound by strict laws of physics. It is a different thing that we cannot measure exactly the various parameters influencing the outcome e.g. the force exerted by the thumb when tossing a coin, the velocity of air, temperature gradients, ballisttics etc. etc. All of these are important, to determine the exact trajectory of the coin, and hence the exact outcome of this experiment. If we could measure (and control) all these, there will be no such thing as a random toss. We can put forth such arguments for all such so-called random phenomena when they involve physical devices and objects.

We will stop tossing coins in the air, or throwing dice on the table, and look for some non-physical phenomenon, for generating random numbers. Let us look at the last digit of the first n (assume n=20 for this case) prime numbers [1]. Here is what we get:
2 3 5 7 1 3 7 9 3 9 1 7 1 3 7 3 9 1 7 1
Looks like a perfect chain of random numbers (perhaps generated
by a ten-faced die). Wrong again -- we can predict exactly the
21st digit which follows the last one above: 3 (the 21st prime
is 73) . Look at this from another viewpoint. Suppose we decide
to write these random numbers in a binary notation. And now, we
look at the last digit only (in this case we must call it a bit).
Here is what we will get:
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ....... ad infinitum.
Nothing could be more predictable than this series !

Now, let us look at the first n (n=20) primes themselves ( The first 1000 primes :: http://www.utm.edu/research/primes ):
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71.... ??

If we know the magic potion which was used for generating the above numbers, then we can predict exactly the number which must follow (the 21st prime number in the series). If we ignore for a moment the monotonically increasing magnitudes, and if we hide the common phenomenon behind these numbers (the fact that they are all primes), we probably have a good candidate for random numbers. Suddenly, the same 21st number becomes random and unpredictable !

The conclusion is simple, there is no such thing as a true random number ! It is just whether we are or we are not able to formalise the underlying generating phenomenon that makes all the difference. Some people prefer to use a more explicit term: non-deterministic phenomenon , to refer to such processes.

Let us go back and look at the question which started all this. The answer is a big NO, because: .... we let the reader work out the arguments. (They rightly say so ...By just looking at the tracks, you can't say which way the train went.)

PS :: If you wish to comment on this, please send me a direct mail : drpartha AT gmail DOT com

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