Winning Probability Maximization Using Random Participant Behavior in the Monty Hall Dilemma

Abstract

In the issue we consider Monty Hall Dilemma. We use decision making approach and computer simulation to solve it. Monty Hall Dilemma is a three step problem involving a participant, a host and three doors with a valuable prize behind one of them and worthless prizes behind two others. The participant should guess where it is to win the valuable prize. After the participant makes an initial choice for one door, the host opens a non-chosen door with a worthless prize behind it. Then the participant is asked whether he wants to stay with his initial choice, or to switch to the remaining unopened door. The problem is quite old and still of much interest for mathematicians and psychologists because of counterintuitive solution and most humans erroneous situation behavior. Traditionally Monty Hall Dilemma is considered as a probability or a game theory problem. We choose an alternative approach and solve it as a decision making task. We determine possible participant’s and host’s actions and their probabilities and construct a problem decision tree. Its leaf nodes represent the situation outcomes. Then we evaluate the outcomes probabilities assuming that the participant sticks to his initial choice with a constant unknown probability. It allowed us to construct a total winning probability function. Its maximization allowed us to determine that an optimal participant behavior is to switch. We also performed computer simulation to verify our theoretical solution.

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