Hello,I am a teacher of Computer Science, from India.

The randomness of randomness

S. Parthasarathy
drpartha@gmail.com

By just looking at the tracks, you can't say which way the
train went..... Anon.

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's 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’s 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: 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. Can someone explain why ? Send your
replies to the author, by email.


References [1] The first 1000 primes
http://www.utm.edu/research/primes


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