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Volume 11: Pages 579-584, 1998

A Coin‐Tossing Experiment that Demonstrates Stochastic Interactions Between Statistically Independent Events

Allen D. Allen

CytoDyn of New Mexico, Inc., 4236 Longridge Avenue, #302, Los Angeles, California 91604 U.S.A.

Consider a large finite permutation C containing #(C) heads and tails produced by randomly tossing a balanced coin. Proceeding in the order of C, construct k ordered x‐tuples, each containing tails uniquely as its last element. It can be shown that at the end of any finite interval of time, the largest x‐tuple will not contain more than approximately −log₂ k⁻¹ elements. Because an approximation of −log₂ k⁻¹ is not strictly monotone increasing as #(C), the following can be demonstrated mathematically and experimentally: When #(C)/2 is large and a developing x‐tuple approaches a length of −log₂ k⁻¹, the empirical relative frequency with which the next coin toss will be heads falls rapidly from 50% to zero. This violates Bayes' theorem, which states that the probability of heads is always 50%. Equivalently, statistically independent measurements can influence one another, extending nonseparability beyond quantum mechanics. The effect can be seen experimentally with the aid of a personal computer to generate large permutations of virtual coin tosses.

Keywords: entanglement, statistical independence, Bayes' theorem, gambler's fallacy, Allen's theorem, standard quantum mechanics, computer experiment

Received: April 3, 1998; Published online: December 15, 2008