THE MONTY HALL PROBLEM

AN INTERESTING PROBABILITY PUZZLE

Background

The Monty Hall problem is a probability puzzle loosely based on the American television game show Let's Make a Deal and named after the show's original host, Monty Hall. The problem, also called the Monty Hall paradox, is a veridical paradox because the result appears impossible but is demonstrably true.

The Problem

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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Common mistake

Most people would suppose that both of the left doors have the same 1/2 chance of winning the car.

Answer

The contestant should switch to the other door. Contestants who switch have a 2/3 chance of winning the car, while contestants who stick have only a 1/3 chance.
Player's pick has a 1/3 chance on the car and 2/3 chance on a goat. As each one has 1/3 chance on the car, the combined chance for door 2 and door 3 to hide the car is 2/3.
With the usual assumptions, after the host deliberately opened a door to intentionally show a goat, but without any further information, the player's pick still retains its 1/3 chance, likewise the other two doors still retain their combined 2/3 chance:
Null for the door opened, but 2/3 for the host's second still unopened door.
The Monty Hall Problem - Explained
The Monty Hall problem has attracted academic interest because the result is surprising and the problem is simple to formulate. Furthermore, variations of the Monty Hall problem are made by changing the implied assumptions, and the variations can have drastically different consequences. For example, if Monty only offered the contestant a chance to switch when the contestant had initially chosen the car, then the contestant should never switch.

Omar Turk & Firas Safa