Monty-Hall (parameterized strategist-host) Theorem: Correcting a Historical Error in Statistical Methodology

Abstract

The Monty-Hall (parameterized strategist-host) Theorem along with a constructive proof is presented, by solving the corresponding Monty-Hall Problem, wherein the host plays a parameterized strategy on the guest.  It establishes the limits on the range of values for the probability of winning the prize.  Eight extreme strategies (corresponding to the set of extreme values for the three perturbation parameters) have been well characterized.  It is shown that there does not exist any strategy 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.  The classical Monty-Hall Problem is a special case with zero-value for each of the three perturbation parameters. 

This paper is an attempt to correct the errors (of long-standing historical significance) in the application of statistical methodology in solving the classical Monty-Hall Problem - one of them being the erroneous use of conditional probabilities for updating the knowledge to facilitate the decision-making by the guest, based on the information about a losing-choice, which itself is dependent on the initial-choice of the guest.  Similar scenarios in data science, machine learning & artificial intelligence can have serious far-reaching consequences.