Interesting comment. I think the statement of the bet implies
that a fair coin is being used. Also, sequences one and
two have both H and T further implying that the coin at
least has two sides. But, in the spirit of your comment,
suppose that the length of the sequence is 50 and the author
claims he gets something like case 3 (all T's) and he insists
the coin is two-sided and fair. I'd be inclined to doubt the
guys integrity :).
This reminds me of a little story in Yockey's book. Two people,
a "true believer" and a skeptic, watch as some great prophet
of the true believers religion tosses a coin. As the coin is
tossed repeatedly the prophet keeps getting heads over and
over. The true believer attributes this to the great power
of the prophet, his faith is greatly increased. The skeptic
attributes it to luck, after all any specific sequence of
tosses has the same probability as any other sequence. When
you really think about it reasonably, the result is not all
that surprising. For some reason, neither asks to see the
coin. :)
But, more to the point, you are exactly right in pointing
out that one has to know the probability distribution
before one can compute a probability. This is in fact one
of the most common mistakes made in probability arguments.
A probability is calculated without even a clue as to
what the probability distribution is. So, thinks for reminding
us of this error.
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"It appears to me that this author is asking
much less than what you are refusing to answer"
-- Galileo (as Simplicio in _The Dialogue_)