### Previous Research on Probabilities  
The research reported in the paper is based on previous research to propose a new methodology to control the nondeterminism of choice operation for smart systems, as shown in Figure 2 with the white block arrows. In previous research, a new process algebra called dTP-Calculus was presented to verify probabilistically the safety and security requirements of the smart systems, with which the nondeterministic behavior of the systems is defined and controlled by the static and dynamic probabilities. In addition, as a result of the verification, in case of the probabilistic requirements not being satisfied, it was shown, as a feasibility ground, that the probabilities making the decisions were enhanced at some arbitrarily high level so that the requirements were to be satisfied. The detailed steps of the approach in the previous research are as follows [11]; note that each step is indicated with the numbers from the figure:  
1. (1-1) The probabilistic OReqs (operational requirements) of smart IoT systems are defined by dTP-Calculus with the ITL (In-The-Large) system model and ITS (In-The-Small) process models. The nondeterminism of the systems is determined by the probabilistic choice operations in the process models.
2. (1-2) A probabilistic EX (execution) model is determined from the ITL model and its ITS models, and all the execution paths are generated. The nondeterministic behavior of the system is constructed by the compositions of the probabilistic choice operations among the ITS models.
3. (1-3) The simulation outcome of each execution path is generated in the form of a GTS (geo-temporal space) model. Each behavior of the system is represented by the composite value of probabilities from the corresponding probabilistic choice operations between the interacting ITS process models.
4. (1-4) The SSReqs (safety and security requirements) are defined by GTS visual logic. Furthermore, the requirements are defined with probabilities too.
5. (1-5) The SSReqs are analyzed and verified on GTSs, especially with respect to the probabilities of the SSReq over the probabilities in the probabilistic EX models.
6. (1-6) In the case where the SSReqs are not satisfied probabilistically, the probabilities of the choice operations in the ITS process models are enhanced to a certain high level, repeating the previous steps until all the SSReqs are satisfied probabilistically.  
The approach in the previous research showed the feasibility of the research goal, which is to define and control nondeterminism in the systems with probability so that the probabilistic requirements can be satisfied nondeterministically in the system. However, the approach reveals the following limitations beyond the feasibility:  
* Continuousness--the value of probability was defined to some value in the continuous range of values between zero and one that is conceptually away from the real situation;
* Arbitrariness--the target value for the new probability to satisfy the requirements was arbitrarily determined without any logical correlation to the physical world.  
In order to overcome the limitations, this paper proposes a new methodology for a set of new notions for probabilistic process and system equivalences, from which the nondeterminism in the systems can be enhanced systematically, that is, discretely and incrementally, until all the SSReqs are satisfied probabilistically.