Monty-Hall (classical-host) Theorem and Monty-Hall (strategist-host) Theorem Dr.(Prof.) Keshava Prasad Halemane, Professor - retired from Department of Mathematical And Computational Sciences, National Institute of Technology Karnataka Surathkal, Srinivasnagar, Mangaluru - 575025, India. https://orcid.org/0000-0003-3483-3521 Independent Researcher (no funding) SASHESHA, 8-129/12 Sowjanya Road, Naigara Hills, Bikarnakatte, Kulshekar Post, Mangaluru-575005. Karnataka State, India https://www.linkedin.com/in/keshavaprasadahalemane/ k.prasad.h@gmail.com ABSTRACT The Monty-Hall (classical-host) Theorem is presented along with a constructive proof by solving the classical Monty-Hall Problem. It establishes the fact that the probability of winning the prize is unaffected by a switched-choice; unlike the most prevalent and widely accepted position held by the leading subject matter experts. A parameterized supermodel is presented, with the associated generic Monty-Hall (strategist-host) Theorem, along with a constructive proof, by solving the corresponding Monty-Hall Problem, wherein the host plays a generic parameterized strategy on the guest. This model subsumes the Monty-Hall (classical) Problem. It establishes the limits on the range of values for the probability of winning the prize, with or without a possible switched-choice. Eight extreme strategies have been identified and characterized. It is established that there does not exist any strategy, that a strategist-host may play on the guest, which would result in a situation wherein a switched-choice will always (irrespective of the placement of the prize and irrespective of the initial-choice of the guest) lead to an enhancement in the chances of winning the prize for the guest. The clearly partitioned three-dimensional discrete event(sample)space, with the twelve mutually-exclusive together-exhaustive possible alternatives, along with the corresponding apriori probabilities, presented as the input data set, is a fail-safe framework to study, analyze & solve the problem; with no possibility of missing any relevant component terms or including any irrelevant component terms, while going through the required calculations in order to derive the desired results. Keywords: Monty-Hall (classical-host) Theorem; Monty-Hall (strategist-host) Theorem; Monty-Hall Theorems; Bayes-Price Rule; Bayes Theorem; Discrete Event (Sample) Space; Parameterized Strategy; Perturbation Parameters; AMS MSC Mathematics Subject Classification: 60A99; 60C99; 62A99; 62C99.