A Lawyer’s Look at Monty Hall

Objection! Assumes facts not in evidence.

Photo by Florian van Duyn on Unsplash

I have had reason lately to think about legal writing. I follow a simple rule there: State all and only the facts that you are going to use, and use all and only the facts that you state. My purpose here is to show how applying that rule to the famous Monty Hall problem would clear the air, but also spoil the fun.

The Problem

Here’s the Wikipedia version of this old chestnut:

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 [whom I call “Monty” below”], who knows what’s behind the doors, opens another door, say №3, which has a goat. He then says to you, “Do you want to pick door №2?” Is it to your advantage to switch your choice?

This statement of “the facts” omits an important fact; it does not tell us under what conditions Monty opens the second door. Maybe he only opens that door if Door №1 hides the car. Or maybe, despite the fact that he knows what’s behind the doors, he nevertheless chooses the doors at random. If either of those possibilities, neither of which is explicitly excluded by the statement of the problem, applies, no one can say whether it is to your advantage to switch.