Question

(Monte Hall Problem). Suppose there are three curtains. Behind one curtain there is a nice prize while behind the other two there are worthless prizes. A contestant selects one curtain at random, and then Monte Hall opens one of the other two curtains to reveal a worthless prize. Hall then expresses the willingness to trade the curtain that the contestant has chosen for the other curtain that has not been opened. Should the contestant switch curtains or stick with the one that she has? If she sticks with the curtain she has then the probability of winning the prize is \(1 / 3 .\) Hence, to answer the question determine the probability that she wins the prize if she switches.

Short answer

The contestant should switch doors. The likelihood to win the prize if she switches is \(2 / 3\), whereas if she sticks with her original choice, the probability of winning is only \(1 / 3\).

Step-by-step solution

Probability of the initial choice

When the contestant makes a choice, there are three curtains, so the probability that she has chosen the one with the prize is \(1 / 3\). The probability that the prize is behind one of the other two curtains is \(2 / 3\).

Revealing a worthless prize

After the contestant makes a choice, the host who knows what's behind each curtain, reveals a worthless prize from the two unselected curtains. This doesn't change the probability of the initial choice, just gives new information.

Calculating the probability if the contestant switches

If the contestant switches, she will win if the prize was behind one of the two unselected curtains, which was the case with the probability of \(2 / 3\) that we calculated in the first step. So, the chance to win after switching is actually \(2 / 3\).

Key concepts

Probability Theory

The Monty Hall problem is a great way to delve into the intricacies of probabilities. Probability theory helps us measure the likelihood of an event occurring. In the context of the Monty Hall problem, we deal with the probability of selecting the correct curtain among the three available ones.
When the contestant first chooses a curtain, the probability of selecting the curtain with the nice prize is \( \frac{1}{3} \). This means there is a 33% chance to guess correctly.

• Initially, one curtain has a prize, and two curtains do not.
• The probability of choosing a curtain with a prize: \( \frac{1}{3} \).
• The probability of the prize being behind one of the other two curtains: \( \frac{2}{3} \).

After one worthless prize is revealed, the contestant's chance of winning by sticking to their initial choice remains at \( \frac{1}{3} \). Meanwhile, a decision to switch curtains changes the probability of winning to \( \frac{2}{3} \). This apparent paradox highlights the counterintuitive nature of probability and is a fascinating introduction to more complex probability theories.

Game Theory

The Monty Hall problem is a classic example within game theory, which studies how different strategies lead to various outcomes based on the actions of others. In this scenario, the contestant is playing against the game host, Monty, who has the inside knowledge of the curtains. Game theory helps understand the decision-making process under strategic situations like these.
The contestant initially picks one option without any knowledge of what lies behind the curtains. Monty's move of revealing a worthless prize introduces a game strategy for the contestant:

• Stick with the initial choice, keeping the winning probability at \( \frac{1}{3} \).
• Switch curtains, altering the chances of winning to \( \frac{2}{3} \).

Game theory shows that Monty's action has shifted the possible outcomes, and the strategic move for the contestant is to switch. This demonstrates how informed strategies can change the likelihood of winning in games.

Decision Making under Uncertainty

One of the most practical applications of the Monty Hall problem is in the sphere of decision making under uncertainty. Decision making under uncertainty involves making informed decisions when not all factors are known or when events are unpredictable.
In the Monty Hall problem, the contestant initially has limited information about which curtain holds the prize. This creates uncertainty about making the correct initial selection. However, once Monty opens a curtain and reveals a worthless prize, the situation changes:

• The initial decision was made with limited information.
• Monty's reveal updates the information set for the contestant.
• The decision to switch offers a calculated move based on the updated probabilities.

This problem emphasizes the importance of reassessing situations when new information is introduced. By switching curtains, the contestant increases her chances of winning, showcasing how to make better decisions with the data available, even when not all elements are initially clear.