At 10:14 AM +0000 2/12/08, Barker, Sean (UK) wrote:
Pat's claim "The definition of a random
sequence is that no matter how
much of it you have, there is no way even
in principle to compute any
information about the next item." is true
only where you exclude
probabilistic estimates (which you might do
depending on how you
interpret "information"). For example, if you encode
the tosses of a
coin as a bit stream, as you continue to observe the bit
stream, you
will be able to make increasing accurate estimates of the
probability
that the next bit will be a 1. Given the additional knowledge
that this
is the encoding of coin flips, you will also be able to
estimate the
probability that it is a fair
coin.
No, wait. A series of tosses of an unfair coin is not a random
sequence. One gets randomness just when the actual probability of each toss
being a head is 0.5 precisely.
What you say above is correct, of course, but it can be translated as: if
a series of bits is not random, this can be detected with increasing accuracy
as the series gets longer. Also, of course, if it is random, this can also be
detected (if it were previously unknown), but that does not mean that any
particular toss can be predicted.
Pat
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