On a paradox of probability theory

R. Jagannathan

Oleg Yu. Vorobyev

Here an improved generalization of Feynman’s paradox of negative probabilities [1, 2] for observing three events is considered. This version of the paradox is directly related to the theory of quantum computing. Imagine a triangular room with three windows, where there are three chairs, on each of which a person can seat [4]. In any of the windows, an observer can see only the corresponding pair of chairs. It is known that if the observer looks at a window (to make a pairwise observation), the picture will be in the probabilistic sense the same for all windows: only one chair from the observed pair is occupied with a probability of 1/2, and there are never busy or free both chairs at once. Paradoxically, existing theories based on Kolmogorov’s probability theory do not answer the question that naturally arises after such pairs of observations of three events: «What is really happening in a triangular room, how many people are there and with what is the probability distribution they are sitting on three chairs?».

Luigi Accardi

Mengxing Wei, Ali al‐Nowaihi, Sanjit Dhami

We report the results of an experiment we performed to test the matching probabilities for the Ellsberg paradox predicted by the quantum decision model of al-Nowaihi and Dhami (2016). We find that the theoretical predictions of that model are in conformity with our experimental results. This supports the thesis that violations of classical (Kolmogorov) probability theory may not be due to irrational behaviour but, rather, due to inadequacy of classical probability theory for the description of human behaviour.

Stanley Gudder

Richard Woesler, Guillaume Adenier, Andrei Yu. Khrennikov +3 more

Extremely many worlds would exist mathematically where the results of quantum processes happen to occur from now on outside earth always according to completely other probability distributions then the known ones. At first glance it seems to be extremely improbable to perceive ourselves in a world where the probability distributions will remain those according to the simple rules of quantum theory. This is called a probability paradox, and further similar paradoxes are presented. A strategy is described to search for solutions, an information theoretic approach. In that approach infinitely many worlds are considered in parallel.