Talk:Impossibility of a gambling system

Definition

"The principle states that it is not possible to select a sub-sequence of a random sequence in a way to improve the odds for a specific event". Really ? I see two problems with this statement:

The sequence must consist of independent events. For example, I could have a sequence of non-independent coin tosses where a each toss has a 75% probability of being the same as the previous toss. By symmetry, the probability that a randomly chosen toss is Heads is 50%, but if I choose the sub-sequence of tosses that follow a Head then the proportion of Heads in this sub-sequence is 75%. In the real world there are workable gambling systems (such as card counting) that exploit non-independence.
More difficult to fix is how to define "select". If I toss a fair coin but "select" a sub-sequence of tosses that are all Heads then I have improved the odds that a randomly chosen toss is Heads from 50% to 100%. If I toss the coin fifty times and notice that there are six Heads between tosses 11 and 20, then I can "select" the sub-sequence of tosses 11 to 20 to improve the odds of Heads from 50% to 60%.

It would be good to see a tighter definition of the principle that avoids these problems. Gandalf61 (talk) 11:24, 9 March 2010 (UTC)[reply]

That was a general definition for the general public at the top. If the sequence has dependent events, how random is it? A tighter, more mathematical def can be added further below. hurch defined select via recur functions. Feel free to read Church's paper and add a formal def therefrom. History2007 (talk) 12:30, 9 March 2010 (UTC)[reply]

Wow - that was a really snippy and defensive response to a polite request. To answer your three points in order:

I am certainly in favour of presenting definitions that are accessible to the general reader, as long as accuracy is not sacrificed. In this case, the definition has been simplified so much that it is no longer correct.
My understanding of "random sequence" in a general context is a sequence that is generated by a stochastic process. This certainly includes sequences in which the terms are not independent - such as Markov chains, for example. If the meaning of "random sequence" in this article is more restricted - if, for example, it means Martin-Löf randomness - then I think this should be made explicit.
A rule that selects tosses that are Heads seems like a perfectly good recursive function to me. But presumably the correct definition of "select" must also restrict the selector's knowledge in some way - perhaps it only allows recursive functions on the previous terms in the sequence. However, I don't have access to Church's paper, so I will have to leave the job of improving this article to yourself and other editors. Gandalf61 (talk) 13:36, 9 March 2010 (UTC)[reply]

Well, if not a recursive function, then maybe a snippy function, I am developing that theory as we speak. But I now clicked on a few articles on randomness, random sequences, algo-random seqs etc. within Wikipedia and it is hard to decide which is worse. Almost all of them contradict each other and all of them have very deterministic and glaring errors. It will take major effort to clean all of this up, and I don't feel like it right now, but later maybe - who knows. History2007 (talk) 16:29, 9 March 2010 (UTC)[reply]