Integrated Circuit (IC) technology has brought great revolution in the field of semiconductors' electronics. The impacts of exemplary usage of IC technology are visible and we realize in our day-to-day life. Varying from highly technical embedded systems like in airplanes, medical robots, traffic controller, bullet trains and other similar, to a common usage systems at home such as smart televisions, intelligent air-conditions, microwaves, washing machines and many more are the results of advances and developments in the field of IC technology. Although design of digital systems and fabrication of ICs have advanced with highly immense pace. However, unfortunately the field of digital system testing has not developed with the same pace. This asynchronization between the pace of design and test developments are due to the main factor, which is "time to market". Due to this the digital system products are coming in the market without following the standard testing procedures, that causes danger to safety and security which turn into great loses of lives and economy. Therefore, this suggests testing the digital systems at its very early stage for rectifying the error. With this aim, the thesis work in the field of testing is chosen for this research work. The work studies the various aspects of testing, surveys the past and current trends. In this thesis some methods towards the optimal solution of testing is proposed. The proposed method uses the concepts of probabilistic modeling controllability, observability and testability. The proposed method is developed in form of an algorithm that provides the information about the weak nodes, After getting this information, the test patterns are generated. The algorithm is applied to numerous circuits to prove the validity of the algorithm and optimal solution of the proposed method.