Economic Classroom Experiments/Monty Hall Paradox

The Monty Hall Paradox is a compelling fallacy in probability where the disclosure of some apparently irrelevant information causes the relative likelihood of two events to change from 1:1 to 2:1. The Wikipedia page has much more information.

Overview

Level

Any level

Prerequisite knowledge

n/a

Suitable modules

Any

Intended learning outcomes

1. Gain an improved understanding of probability and Bayes' rule.
2. Understand the origins of the paradox.

Computerized version

A computerized version of this experiment is available on the Exeter games site.

You can quickly log in as a subject to try out this individual progress experiment. You may also find the sample instructions helpful.

Abstract

Students play multiple rounds individually against the computer and in each round try to locate a prize that has been hidden at random in one of three closed boxes. The student starts by guessing where the prize is, after which the computer opens one of the other two boxes that is empty and gives the student the option of sticking with his/her original choice or changing to the remaining unopened box. This game is the well-known Monty Hall game show paradox where the student is the contestant and the computer is Monty.

Changing is twice as likely to be successful as sticking. Because this is so counter-intuitive, the instructor may also configure a repeat-play 'strategy' version of the game, where the student plays the game once in each round as before but then the computer plays the game a further large number of times using the student's choice of initial box and 'strategy' of 'stick' or 'change'. The computer displays the results of the individual games plus a summary.

There are also versions of the game with four and five boxes.

Discussion of how the paradox arises in the three-box game

The original guess plainly has a 1/3 probability of being correct. The paradox arises principally because it does not seem as if the computer is imparting any useful information by opening an empty box. However there is a 2/3 probability of the prize being in one of the two boxes that were not guessed, so after the computer helpfully eliminates one of them, the 2/3 probability remains attached to the remaining unopened box, which is therefore twice as likely to contain the prize.