paradoxes of randomness

[Tim May]

On Tuesday, August 19, 2003, at 03:13  AM, Sarad AV wrote:
>> In a perfectly random experiment,how many tails and
>> how many heads do we get?
> we don't know - or it wouldn't be random :)
>> for a sufficiently large sample you *should* see
>> roughly equal numbers of heads and tails in the
>> average case.
>
> We say that, we-don't know or it wont be random. Then
> we say that we must see roughly equal numbers of heads
> and tails for large trials. Thats what I fail to
> understand.

Start small. Do some experiments _yourself_.

Take a coin out of your pocket. I assume your local coin has something 
that may be called a "head" and something that may be called a "tail." 
In any case, decide what you want to call each side.

Flip the coin very high in the air and let it land on the ground 
without any interference by you.

This is a "fair toss." (That subtle air currents may affect the landing 
is completely unimportant, as you will see even if you have doubts 
about it now.)

Now let's try a little piece of induction on this one, single toss. 
Remember when you had said earlier that a "perfectly random coin toss" 
would have exactly equal numbers of heads and tails? Well, with a 
single toss there can ONLY be either a head or a tail.

The outcome will be ONE of these, not some mixture of half and half.

This proves, by the way, that any claim that a random coin toss must 
result in equal numbers of heads and tails in any particular experiment.

Now toss the coin a second time and record the results.

(I strongly urge you to actually do this experiment. Really. These are 
the experiments which teach probability theory. No amount of book 
learning substitutes.)

So the coin has been tossed twice in this particular experiment. There 
is now the possibility for equal numbers of heads and tails....but for 
the second coin toss to give the opposite result of the first toss, 
"every time, to balance the outcomes," the coin or the wind currents 
would have to "conspire" to make the outcome the opposite of what the 
first toss gave. (This is so absurd as to be not worth discussing, 
except that I know of no other way to convince you that your theory 
that equal numbers of heads and tails must be seen cannot be true in 
any particular experiment. The more mathematical way of saying this is 
that the "outcomes are independent." The result of one coin toss does 
not affect the next one, which may take place far away, in another 
room, and so on.)

In any case, by the time a third coin toss happens there again cannot 
be equal numbers of heads and tails, for obvious reasons. And so on.

Do this experiment. Do this experiment for at least 10 coin tosses. 
Write down the results. This will take you only a few minutes.

Then repeat the experiment and write down the results.

Repeat it as many times as you need to to get a good feeling for what 
is going on. And then think of variations with dice, with cards, with 
other sources of randomness.

And don't "dry lab" the results by imagining what they must be in your 
head. Actually get your hands dirty by flipping the coins, or dealing 
the cards, or whatever. Don't cheat by telling yourself you already 
know what the results must be.

Only worry about the deep philosophical implications of randomness 
after you have grasped, or grokked, the essence.

(Stuff about Kripke's possible worlds semantics, Bayesian outlooks, 
Kolmogoroff-Chaitin measures, etc., is very exciting, but it's based on 
the foundations.)


--Tim May

"We should not march into Baghdad. To occupy Iraq would
instantly shatter our coalition, turning the whole Arab
world against us and make a broken tyrant into a latter-
day Arab hero. Assigning young soldiers to a fruitless
hunt for a securely entrenched dictator and condemning
them to fight in what would be an unwinable urban guerilla
war, it could only plunge that part of the world into ever
greater instability."
--George H. W. Bush, "A World Transformed", 1998