The Monty Hall Problem
When Should We Change Our Mind?
A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams, Chapman & Hall/CRC Press. 2024.
THE GENESIS OF THE MONTY HALL PROBLEM
The Monty Hall Problem was named after the original host of the American game show, ‘Let’s Make a Deal’. It became a topic of popular debate because of the answer provided to a question quoted in a column in Parade magazine.
The concept is that contestants are given a choice of three doors. Behind one door lies a highly desirable prize like a car, while behind the other two doors were much less desirable prizes like goats. The car is placed randomly behind one of the doors, preventing contestants from predicting its location based on prior observations or information.
THE PUZZLE UNVEILED
Imagine yourself on this game show. You are asked to choose one of three doors (let’s call them Doors 1, 2, and 3). After making your choice (let’s say you choose Door 1), the host, who knows what’s behind each door, opens another door (for instance, Door 3) to reveal a goat.
He then offers you a choice. You can stick with your original decision (Door 1 in this case), or you can switch to the remaining unopened door (Door 2). You should note that the host always opens a door that you didn’t choose and that hides a goat, increasing the suspense and making the game more interesting.
The question that the Monty Hall problem asks is: Should you stick with your original choice, or should you switch to the other unopened door?
THE COUNTERINTUITIVE ANSWER
At first glance, it might seem like your odds of winning the car are the same whether you stick to your original choice or switch. After all, there are only two doors left unopened, so isn’t there a 50% chance that the car is behind each of them?
In her column, Marilyn Vos Savant argued that the chance is not 50% either way, but that you have a higher chance of winning the car if you decide to switch doors. Despite receiving numerous objections from readers, including some leading academics, her answer holds up under scrutiny. Here’s why.
When you first choose a door, there is a 1 in 3 chance that it hides the car. This means that there’s a 2 in 3 chance that the car is behind one of the other two doors. Even after the host opens a door to reveal a goat, these probabilities do not change. Monty is simply providing more information about where the car is not.
So, if you stick with your original choice, your chances of winning the car remain at 1 in 3. However, if you switch, your chances increase to 2 in 3. Switching doors effectively allows you to select both of the other doors, doubling your odds of finding the car.
A CLOSER LOOK AT THE PROBABILITIES
Let’s examine the situation more closely to understand how this works.
If the car is behind Door 1, and you choose it and stick with your choice, you win the car. The chance of this happening is 1/3.
If the car is behind Door 2 and you initially choose Door 1, the host will open Door 3 (since it conceals a goat). If you switch to Door 2, you win the car. The chance of this happening is 2/3.
If the car is behind Door 3, and you initially choose Door 1, the host will open Door 2 (since it conceals a goat). If you switch to Door 3, you win the car. The chance of this happening is also 2/3.
From the above, you can see that you have a 2/3 chance of winning if you switch to whichever door Monty has not opened, and a 1/3 chance of winning if you stick to your initial choice.
THE ROLE OF THE HOST
It’s crucial to note that the host’s knowledge and actions play a pivotal role in these probabilities. If the host didn’t know what was behind each door or randomly chose a door to open, then the odds would indeed be 50–50, as he might have inadvertently opened a door to reveal the car. However, because the host always opens a door you didn’t choose and always reveals a goat, the odds shift in favour of switching doors.
To expand upon this, consider a version of the problem with 52 cards. This time, you’re invited to choose one card from a deck of 52. The objective is to select the Ace of Spades from a deck of cards lying face down on the table.
If you initially choose the Ace of Spades and stick with your choice, you win the game. The chance of this happening is 1/52, since there’s only one Ace of Spades in a 52-card deck.
However, if you initially choose any card other than the Ace of Spades (which has a 51/52 chance), the host, knowing where the Ace of Spades is, will begin to turn cards over one at a time, always leaving the Ace of Spades and your initial card choice in the remaining face-down deck. The host will continue to do this until only your card and one other card remain. One of these two cards will be the Ace of Spades.
At this point, there is still a 1/52 chance that your original card is the Ace of Spades. If you switch your choice to the remaining card, the chance that it will be the Ace of Spades is therefore 51/52, which is a much higher probability than if you stick with your initial choice.
This works because the host each time deliberately turns over a card that is not the Ace of Spades. So the other card left face down at the end is either the Ace of Spades, with a chance of 51/52, or else your original choice is the Ace of Spades, with a probability of 1/52.
If the host doesn’t know where the Ace of Spades is located, he might inadvertently reveal it every time he turns a card over, so he would be providing no new information about the location of the Ace of Spades by exposing a card.
This shows how the Monty Hall problem can scale to larger numbers. The initial odds of choosing the Ace of Spades are 1/52, but if you switch your choice after the host takes away all but one of the other cards, your odds improve dramatically to 51/52. This is a counterintuitive result, but it follows from the fact that the host’s actions (because he knows where the Ace of Spades is) give you additional information about where the Ace of Spades is not.
OVERCOMING INTUITION WITH LOGIC
The Monty Hall problem can be difficult to grasp because it seems to contradict our intuition. The human brain tends to simplify complex situations, and when there are two unopened doors, it’s easy to fall into the trap of assuming there’s a 50% chance of winning either way. However, the Monty Hall problem highlights how understanding probability requires careful thought and a logical analysis of the situation.
EXPLORING THE MONTY HALL PROBLEM WITH SIMULATIONS
If you’re still having trouble grasping the Monty Hall problem, you might find it helpful to see it in action. Numerous online simulators let you play the Monty Hall game repeatedly, and over time, you’ll see that switching doors indeed wins about 2/3 of the time.
THE MONTY HALL PROBLEM IN POPULAR CULTURE
The Monty Hall problem has seeped into popular culture, appearing in films, television series, and even songs. It serves as a reminder that intuition and probability sometimes have a complicated relationship. The logical and statistical reasoning involved in this puzzle, as well as its seemingly paradoxical result, have made it a favourite topic in probability and statistics classes across the world.
CONCLUSION: PROBABILITY AND INTUITION
The Monty Hall problem is a captivating illustration of how probability can sometimes be counterintuitive. Although it’s been debated, analysed, and confirmed many times over, it continues to intrigue and perplex. It provides a clear lesson: intuition isn’t always reliable when it comes to probability.