
Re: say I flip a coin 100 times...
Posted:
Apr 17, 2005 1:20 PM


HERC777 wrote:
> Right! > > So if each flipper represented a real number, and Herc > starts flipping/making the antidiagonal (#1) at t=1, t=2, > ...
There are two main cases:
1. Herc is flipping a fair coin, without looking at other results. In this case, the probability of matching the first n elements of the antidiagonal is 2^{n}, which tends to zero as n tends to infinity.
In this case, Herc's infinite sequence may or may not match some other coin tosser's sequence, but it has zero probability of being any particular sequence, including the antidiagonal.
2. Herc is looking at the diagonal, and controlling the coin to ensure his tosses form the antidiagonal, breaking the assumption that the coin tosses are independent events. The probability that Herc's first n tosses match any of the first n sequences is zero, for all n, so the limit as n tends to infinity is also zero.
In this case, Herc must form a new sequence, because we know his sequence is different in at least one place from each of the other coin tosser's sequences.
As a metacomment, to prove countability of the set of all coin toss result sequences it would be insufficient to find an error in the diagonal proof. That would merely shift the issue to being unknown. One would have to positively prove that there exists a function F from the natural numbers into the set of all coin toss sequences such that each coin toss sequence is F(n) for some natural number n.
Patricia

