we don't know - or it wouldn't be random :)In a perfectly random experiment,how many tails and how many heads do we get?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
