Quandaries and Queries
First let me express my appreciation for your valuable and admirable service.
I am a screenwriter, currently in the fortunate position of having the development of a Screenplay funded by the South Australian Film Corporation. The (anti)hero of this screenplay is a statistician whose life is falling apart around him, thanks in part to his obsession with perfecting an "ideal" gambling system. Obviously the constraints of film prevent the development and presentation of a sophisticated gambling system, but I would like to provide the audience with enough of its features to give the impression that they are components of credible (though flawed) system. It would be nice to base the system on a very simple, intuitively accessible mathematical concept, and after some reflection I though the golden ratio"phi" might be a good candidate, however this is only an vague and intuitive notion: really I have no clear conception of how "phi" applies in probability and statistics. So my three part question is:
I understand that this question lies outside the domain of typical questions posed to you, and I`ll understand if its irregularity dissuades you from answering it. Any response would be gratefully received!
Thanks very much,
This question is very interesting and would recommend to use the notion of MARTINGALE. Among gamblers, a martingale is the name of a certain strategy which some optimistic gamblers believe is a sure winner. Suppose that you are planning on a roulette: If you bet on red, you get twice your stake if red comes our, and you lose your stake if not. The martingale strategy goes as follows. You start betting say $1 on red; if you lose you bet $2 on red: if you lose again you bet $4 etc., doubling the stake whenever you lose until you finally win. Then you start all over again betting $1 on red, etc. Observe that if red comes out the nth game, then your strike was 2n-1 and you win $2n. Moreover, the sum of your stakes in the n games equals: 1+2+4+...+2n-1=2n -1. Thus, your net profit is increased by $1 whenever red comes out.
The martingale strategy seems to a sure way to beat the gambling house - but be careful! It is instead a sure way to be ruined. The probability of red in different roulettes are:
European roulette: 18/37=48.6%
In all cases it is less than 1/2=50%. According to the famous Optional Sampling Theorem (very difficult even to formulate!) if you will play so-called optional strategy, then the longer you play the more you loses. Thus, the only way to beat the odds is to use non-optional straggles. But then your loss becomes unsoundly large. You can't beat the odds!!
If you want, I can find some references in literature (especially Russian: Dostoevsky and Kuprin), the martingale strategy is very famous and was used by centuries.
As far as I know, the Golden Ratio is NOT used in gambling. The only result I know about applications of the Golden Ratio in Probability Theory is my short paper
ON THE GOLDEN RATIO AND THE STRONG LAW OF LARGE NUMBERS FOR SUMS OF CORRELATED RANDOM VARIABLES WITH APPLICATION TO FIRST PASSAGE TIMES, by Tien-Chung Hu, Andrew Rosalsky, and Andrei I. Volodin, to appear in Mathematical Scientist.
If you want, I can send you a copy (PDF file by e-mail), while it does not discuss gambling questions.
With kind regards,