Monty Hall Problem

The Illusion of Choice

One of the most famous television game shows from the 1950s to the 1980s was “Let’s Make a Deal.” Its host, Monty Hall, achieved a second kind of fame when a probability theory dilemma, loosely based on the show, was named after him. The Monty Hall problem is a well-known, seemingly paradoxical problem where information affects your decision, even though it initially appears as though it shouldn’t.

Imagine you’re on a show, asked to choose between three doors. Behind two doors are goats, and the third door hides a brand-new car. You make your choice. The host, Monty Hall, who knows what’s behind each door, opens one of the other doors to reveal a goat. By the rules of the game, Monty always reveals a goat. He then asks if you want to switch your choice to the other remaining door. What should you do?

In 1990, Marilyn vos Savant addressed this problem in her column in Parade Magazine, a supplement in the Sunday edition of hundreds of American newspapers. Known as the world’s smartest woman for her Guinness World Record for the highest IQ score, vos Savant advised that you should switch doors. She explained that the odds of a car being behind the other door are two in three, while the odds of it being behind your original choice are one in three.

Her column received ten thousand letters, including a thousand from PhDs in mathematics and statistics, many claiming she was wrong. A common response was that there is no difference between switching and not switching. They argued that since one of the two doors not chosen originally had a goat, Monty’s revealing a goat provided no new information. With two doors left, they believed the probability of a car being behind each door was now fifty percent.

However, this argument is incorrect. The correct strategy is to switch, doubling the odds of winning a car. Let’s demonstrate why.

To understand the Monty Hall problem, let’s analyze the three possible locations for a car and evaluate the outcomes based on different strategies.

When you make your first choice, a car could be behind any of the three doors, giving you a one-in-three chance of selecting a car and a two-in-three chance of selecting a goat. Monty then opens a door to reveal a goat, based on the rule that he must always show a goat and never a car.

Suppose you pick Door 1. If you choose to “Stay” with your initial pick:

  1. If a car is behind Door 1, you win.

  2. If a car is behind Door 2, you lose.

  3. If a car is behind Door 3, you lose.

Thus, the probability of winning by staying with your initial choice is one in three.

Now, let’s consider the “Switch” strategy:

  1. If a car is behind Door 1, you lose because you switch away from a car.

  2. If a car is behind Door 2, Monty opens Door 3, revealing a goat, so you switch to Door 2 and win.

  3. If a car is behind Door 3, Monty opens Door 2, revealing a goat, so you switch to Door 3 and win.

The probability of winning by switching is two in three, which is double the probability of winning by staying. Therefore, the optimal strategy is always to switch. Statistically, over many iterations of the game, you will win approximately two-thirds of the time if you consistently switch doors.

This counterintuitive solution highlights the importance of understanding probability and demonstrates how new information can significantly alter decision-making outcomes. The Monty Hall problem remains a fascinating example of how intuition can often lead us astray in probabilistic reasoning.

Words of wisdom

“Our wounds are often the openings into the best and most beautiful part of us.” —David Richo

“I wanted to change the world. But I have found that the only thing one can be sure of changing is oneself.” ―Aldous Huxley

“Judge a man by his questions rather than by his answers.” —Voltaire

“Sometimes people don’t want to hear the truth because they don’t want their illusions destroyed.” —Friedrich Nietzsche

Bibliography

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