ABSTRACT
We now come to the question of deducing the law of excluded gambling systems. Roughly this law may be stated as follows: Suppose we have a probability system (https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203828793/ecd34538-6a11-4738-a799-0c17881c8935/content/inline-2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> s , Ω, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203828793/ecd34538-6a11-4738-a799-0c17881c8935/content/inline-3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>,p)) and suppose A ∈ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203828793/ecd34538-6a11-4738-a799-0c17881c8935/content/inline-2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with p(A) = p. Event A has probability p in a sequence of repetitions of https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203828793/ecd34538-6a11-4738-a799-0c17881c8935/content/inline-2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> s and the lawstates that we cannot increase our chance ofgetting A by selecting a subsequence from the sequence of repetitions. To get a more precise statement let us introduce a sequence of random variables ξ l,ξ 2,…, ξn,… where ξn = 1 if A occurs at the nth repetition, and = 0 otherwise. Bythe axiom of independent repetitions these random variables are independent and have thesame distribution, namely p(ξn = 1) = p, p(ξn = 0) = 1 — p where 0 ≤ p ≤ 1. A gambling system (using Church's approach) is an effective algorithm enabling us to calculate a value gi = 1 or 0 given i and the results of ξ 1,…, ξi –1. If g i = 1 we bet that the ith trial will give A. If gi = 0, we do not bet. Suppose we employ the gambling system on a finite initial segment n of our sequence of repetitions. Let the set In = î (g = 1 and i ≤ n) contain n’ members. Let the number of ξi where i ∈ In and which give an observed value of 1 be m’. Then if m’/n’ > pthe gambling system will have been successful for this initial segment. A gambling system can be said to be successful in general if for all or at least most large n’ it is successful.
