paradoxes of randomness

[Tim May]

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

> hi,
>
> Hope you can help on this.
>
> --- Tim May <timcmay at got.net> 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