The Monty Hall Problem Explained

by | Nov 5, 2018

You chose Door No. 1 originally. Now that door No. 3 has a goat, should you switch?

The “Monty Hall Problem” is a mathematical brain teaser. It is called the “Monty Hall Problem” because it sounds like a question that would be on the game show Let’s Make a Deal which was hosted by Monty Hall. The Monty Hall Problem was submitted as a question to the Parade Magazine “Ask Marilyn” column in 1990. “Marilyn” is Marilyn vos Savant, who was listed as having the highest recorded IQ in the Guinness Book of World Records.

Here’s the question:

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?

Marilyn explained in her column that you should switch doors. If that seems incorrect you are not alone as over 90% of the reader mail Marilyn received disagreed with her, including people with math PhDs! However, Marilyn is correct, the probabilities are better if you switch doors. In her book The Power of Logical Thinking, Marilyn said that the Monty Hall Problem “itself is not difficult to grasp, but because the intuitive answer seems so obvious – and that ‘obvious’ answer is dead wrong – it ensnares many people” because when an answer seems obvious people tend to jump to an early conclusion and it is hard to go back and think through it.

Here’s a simple explanation of why you should switch your door selection:

  1. Before any doors are opened, there is a 1/3rd chance of the car being behind any one of the doors.
  2. Similarly, there is a 2/3rd chance that the car is behind any two of the doors. This is the key point that we fail to realize!
  3. So, when you’ve selected door No. 1, you have a 1/3rd chance of winning and doors 2 and 3 have a 2/3rd chance of winning.
  4. When the host opens door 3 to reveal a goat, we’re still in the same situation: Doors 2 and 3 have a 2/3rd chance of winning and door 1 has a 1/3 chance of winning.
  5. Thus, when the host asks if you want to switch you should say yes. That is because your door, No. 1, has a 1/3rd chance of winning and Door No. 2 has a 2/3rd chance of winning.

Here’s another schematic:



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