The Philosophy of Probability Value Behavior: Fractions and Composite Probability Functions in the Continuous Case

This paper focuses on probability value behavior in the case of continuous sample space by employing fractions intervals and composite functions. The study evaluates value behavior rather than finding values directly, which involves utilization of some concepts from continuity, geometric probability, and measure theory. This paper primarily uses an experiment that contains two major events, head H and tail T, in all their occurrence phases. This spread in infinite and uncountable fractions by a continuous motion within intervals and in the predominant circumstances where events are probabilistic values. As a result, every circumstance reflects many important characteristics of probability theory. Among the main results, this paper provides proven propositions that help design experiments upon understanding the case nature, with some explanations to the existing relation between probability value and the case nature. Also, this paper provides a proven corollary that allows visualizing negative probability values as a particular trial. This in turn proposes necessary uses for the composite probability function P_j 〖(p〗_i). Moreover, this paper provides numerical explanations of limits, which can demonstrate the nature of P_j 〖(p〗_i), alongside some techniques. Also, this paper considered conditional probability through some corollaries and the possibility of using the non-negative function of the interval i, alongside many important results in form of discussions.