The Monty Hall Problem (1 of 3): Causal Bayesian Network
“Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, ‘Do you want to pick door #2?’ Is it to your advantage to switch your choice of doors?”
Pearl, Judea. The Book of Why: The New Science of Cause and Effect (p. 190). Basic Books. Kindle Edition.
I implemented this game as a Causal Bayesian Network (arcs represent causal relationships). Three nodes describe this domain, each one with one state per door:
- Your Door: the initial choice of the player. We assume a uniform prior distribution;
- Location of Car: the door behind which is the car. We also assume a uniform prior distribution;
- Door Opened: this is the door Monty Hall opens based on the following two rules:
- do not open the door that has been chosen by the player;
- do not open the door behind which the car is.
We can interpret the first row of this conditional probability table as "If the player chose Door #1 and the car is behind Door #1, then Monty can open either Door #2 or Door #3". As per the second row, the interpretation is: "If the player chose Door #1 and the car is behind Door #2, then Monty can only open Door #3".
You can download the attached XBL file and open it with any version of BayesiaLab.
You can also experiment with this model via our WebSimulator: https://simulator.bayesialab.com/#!simulator/137248128314