In message , HAVRILIAK
writes
TO: Guy Macon
Thank you very much for your comments. I've tried your suggestion to prove the
point. Instead of flipping a coin I did the following calculation. If you
flip an unbiased coin 2 million times it will come up heads 1 million times and
tails 1 million times. Thus there will be 1 million moves in the positive
direction and 1 million moves in the negative direction.

Your calculation is based on a wrong assumption. I suggest you *really*
go and try this yourself with a genuine coin (probably very slightly
biassed). There is no substitute for doing the experiment here.

You should only need to do it 20 times to see that it is rather unlikely
that you get a result of *exactly* 10 Heads and 10 tails every time. Try
it a second time and you will get a different result.

For the situation where p(heads) = p(tails) = 0.5 you can fairly easily
compute the probability of getting a run of 10 heads, right through to a
run of 10 tails. The same applies to the case of 2 million trials. It
would be very rare to see *exactly* 1 million of each outcome.

The majority of the outcomes will be symmetrically distributed round
1000000 +/- 1000 with extended tails stretching out to either side.

Excel users can simulate the task easily enough by summing a column
containing the expression

if(rand()0.5,-1,1)

f9 to recalc and you will see a series of different answers each time.

Since the moves are
commutative, i.e. a+b=b+a, the net result is no shift. Now if we bias the coin,
there will be a drift in one direction or another depending on the bias.

If you were right the universe would be a very dull place.

Among the first people to record detailed experiments on real (near
perfect 6 sided dice) was a Swiss astronomer called Wolf in the late
1800's. He did find manufacturing bias in the die as well as verifying
modern statistics.

Regards,
--
Martin Brown