Question

Monty Hall's dilemma: a game show problem. You are a contestant on a game show. There are three closed doors: one hides a car and two hide goats. You point to one door, call it \(C\). The gameshow host, knowing what's behind each door, now opens either door \(A\) or \(B\), to show you a goat; say it's door \(A\). To win a car, you now get to make your final choice: should you stick with your original choice \(C\), or should you now switch and choose door B? (New York Times, July 21, 1991; Scientific American, August 1998.)

Short answer

You should switch to door B because it gives you a 2/3 chance of winning the car, compared to a 1/3 chance if you stay with door C.

Step-by-step solution

Understand the problem

You initially choose one of three doors (door C). A goat is revealed behind one of the other two doors you didn't choose (let's say door A). You must decide whether to stick with your original choice (door C) or switch to the remaining door (door B) to maximize your chances of winning the car.

Calculate the initial probabilities

When you first choose a door, you have a 1/3 chance that the car is behind your chosen door (door C) and a 2/3 chance that the car is behind one of the other two doors.

Consider the host's action

After you choose door C, the host reveals a goat behind one of the remaining doors, say door A. This changes the information you have and the probabilities of where the car might be.

Calculate probabilities if you switch

If the car was behind door A or door B (with initial probability 2/3), the host will reveal a goat behind the other door, leaving the car behind the remaining door (here, door B). So, if you switch to door B, you have a 2/3 chance of winning the car.

Calculate probabilities if you stay

If you stay with your original choice (door C), your probability of winning the car remains 1/3 because your initial choice would only be correct 1/3 of the time.

Conclusion

Switching to door B gives you a higher probability of winning the car (2/3) compared to sticking with your original choice, door C (1/3). Therefore, you should switch to door B to maximize your chances of winning the car.

Key concepts

Probability Theory and the Monty Hall Problem

Probability theory helps us make sense of uncertain events and calculate the likelihood of different outcomes. In the Monty Hall problem, we encounter a scenario with three doors. Initially, each door has an equal probability of hiding the car: 1/3. Once the host reveals a goat, these probabilities change based on new information.
When you first choose a door, the car could be either behind the door you picked (1/3 probability) or one of the other two doors (2/3 probability). The host, knowing where the car is, always reveals a goat, and this action helps us update our probabilities.
Switching doors effectively means embracing the higher probability. If you switch, you rely on the initial higher likelihood that the car is behind one of the other two doors (2/3). Therefore, you increase your chances of winning by choosing the door the host did not reveal.

Game Theory Insights

Game theory studies strategic decision-making, where outcomes depend on the choices of all participants. The Monty Hall problem is a perfect example, involving you and the host.
The host's actions are crucial within the game. By revealing a goat, they provide valuable information about where the car is more likely to be. As a contestant, your strategy should adapt to this information. Game theory tells us to maximize payoffs and minimize risks. Here, by switching doors, you capitalize on the changing probabilities, boosting your odds of winning from 1/3 to 2/3.
In essence, understanding the host's intentions and reactions is key in game theory. Knowing the host will always reveal a goat makes switching a strategic move, aligning with game theory principles to maximize success in uncertain conditions.

Decision Making Under Uncertainty

In many situations, including the Monty Hall problem, we face decisions without certainty. Decision-making under uncertainty involves analyzing risks and potential outcomes. Initially, it might seem your chances of winning are equal, whether you stay or switch. However, deeper analysis reveals switching significantly increases your odds.
Here's why: When you chose door C initially, there was a 1/3 chance of picking the car and a 2/3 chance it was behind doors A or B. When the host shows a goat behind door A, the situation changes. Now, with door A eliminated, the 2/3 probability that the car was not behind your initial choice shifts to door B.
Effective decision-making leverages this updated information. By choosing to switch, you act on the improved chances of winning. Evaluating and responding to new data—like the host's reveal—demonstrate good decision-making skills in the face of uncertainty.