hi, Hope you can help on this. --- Tim May wrote:

I hope you are not saying that you think there will always be 16 heads and 16 tails!

In a perfectly random experiment,how many tails and how many heads do we get? thanks. Regards Sarath. __________________________________ Do you Yahoo!? Yahoo! SiteBuilder - Free, easy-to-use web site design software http://sitebuilder.yahoo.com

hi, Thank you-one more question. Will the information obtained from the 2^32 tests have a zero compression rate? If one of the occurance should yield all heads and one occurance yields all tails-there appears to be scope for compression. If the output is random,then it will have no mathametical structure,so I shouldn't be able to compress it at all. Regards Sarath. --- Dave Howe wrote:

for a sufficiently large sample you *should* see roughly equal numbers of heads and tails in the average case - but : for 32 coins in 2^32 tests you should see: one occurance of all heads (and one of all tails) 32 occurances of one tail, 31 heads (and 32 of one head, 31 tails) 496 occurances of two and so forth up the chain none of these are guaranteed - it *is* random after all - but given a sufficiently large number of tests, statistically you should see the above.

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Tim May wrote:

But, as I said in my last post, before you try to understand algorithmic information theory, you need to learn the basics of probability. Without understanding things like combinations and permutations, binomial and Poisson distributions, the law of large numbers, standard deviations, etc., your philosophizing will be ungrounded.

A good place to learn the basics: http://web.jfet.org/6.041-text/ -- Riad Wahby rsw@jfet.org MIT VI-2 M.Eng

(Will whomever prepending this "Re: *** GMX Spamverdacht ***" header please STOP.) On Monday, August 18, 2003, at 09:01 AM, Dave Howe wrote:

randomness is a funny thing. a truely random file can be *anything* that is the right length - including a excerpt from william shakespere (provided you have enough monkeys) it is most likely to be random garbage, for a large sample - but for a very small sample the odds of it being an ordered sample are surprisingly good.

the obvious example here is a double coin test - two bits. an ordered sample would be both heads or both tails. a disordered sample would be one head and one tail. in practice, you would expect to see half the results from a larger trial (say 32 double-throws) with a ordered sample, and half disordered. as you reach three coins, the odds of a completely ordered result decrease (from 2 in 2^2, to 2 in 2^3). for four coins, you still have the same two compressable results. consider: HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH TTTT

in a large trial, you would expect to see each of these once every 2^4 tests, on the average. obviously HHHH and TTTT are very compressable. assuming a runlength encoding, I don't think any of the others are compressable at all...."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

Quibble: only surprising if one misunderstands probability. (Not saying you do, just quibbling with any claim that readily calculated probabilities can be "surprising.") 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 For any sequence of n fair tosses, the "length after compression" of the outcome is roughly, modulo some constants and factor, (1 - 2^(-n)), where "1" is the uncompressed length. In other words, as n gets large nearly all strings have a "length after compression" that is close to the original length, i.e., little compression. As n gets arbitrarily large, an arbitrarily small fraction of strings have a short, concise description (are "compressed"). --Tim May

hi, --- Dave Howe wrote: . (Not

...

saying you do, just quibbling with any claim that readily calculated probabilities can be "surprising.") I meant surprising for Sarad - Much of this discussion pre-assumes that he *does* misunderstand probability but is willing to substitute our

collective insanity for his current ignorance :)

No more of that-I will have a good read. I am basically confused of the fact

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. The idea of a perfect random experiment was taken just to understand the concept. Thanks. Regards Sarath. __________________________________ Do you Yahoo!? Yahoo! SiteBuilder - Free, easy-to-use web site design software http://sitebuilder.yahoo.com

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

On Monday, August 18, 2003, at 03:24 AM, Sarad AV wrote:

hi,

Hope you can help on this.

--- Tim May wrote:

...

I hope you are not saying that you think there will always be 16 heads and 16 tails!

In a perfectly random experiment,how many tails and how many heads do we get?

First, those who think there are "perfectly random" experiments or numbers are living in a state of sin, to paraphrase John von Neumann. Second, despite the above, good approximations to random behavior are easy to obtain: radioactive decays, lava lamps, Johnson noise in diodes, etc. The important aspects of probability theory emerge even with "imperfect" sources of apparent randomness. Third, the expected distribution of heads and tails in a series of 32 coin tosses is approximated closely by the binomial distribution. The key concepts are combinations and permutations. The expected probability of "all heads" is given by (0.5)^32. There are more chances of seeing 1 head and 31 tails, as the head can appear in any of the 32 positions. ("the combination of 32 things taken 1 at a time"). And so on, up to the maximum of 16 heads, 16 tails...although this particular outcome is not very likely. Fourth, my point was that there is relatively low probability that 32 tosses will result in exactly 16 heads and 16 tails. Given enough experiments, the distribution of outcomes will approximately follow the familiar bell-shaped curve, centered at 16H/16T, but with some chance of each of 0H/32T, 1H/31T,...., 31H/0T, 32H/0T. Fifth, not to sound harsh or snippy or sarcastic, but this is really basic stuff. There is a big gap in your education. Even if not taught this in 9th grade (or whatever the equivalent is in India), this is stuff that should be apparent through thinking about the way chance events occur. I urge you to suspend your "advanced math" questions until you have gone back over the more basic things. (It's crazy to try to understand entropy and the algorithmic information theory work of Greg Chaitin and others without knowing the 18th-century results on probability of people like Pascal, Poisson, etc.) There are many Web pages with class notes on probability, encyclopedia entries, etc. And I suggest experiments with coin tosses. And cards. And dice. --Tim May "A complex system that works is invariably found to have evolved from a simple system that worked ...A complex system designed from scratch never works and cannot be patched up to make it work. You have to start over, beginning with a working simple system." -- Grady Booch