Question

A television show features the following weekly game: A sports car is hidden behind one door, and a goat is hidden behind each of two other doors. The moderator of the show invites the contestant to pick a door at random. Then, by tradition, the moderator is obligated to open one of the two doors not chosen to reveal a goat (there are two goats, so there is always such a door to open). At this point, the contestant is given the opportunity to stand pat (do nothing) or to choose the remaining door. Suppose you are the contestant, and suppose you prefer the sports car over a goat as your prize. What do you do? (Hint: It may help to model this as a two-stage dependent trials process, but it may not be obvious how to do this). (a) Suppose you decide to stand with your original choice. What are your chances of winning the car? (b) Suppose you decide to switch to the remaining door. What are your chances of winning the car? (c) Suppose you decide to flip a fair coin. If it comes up heads, you change your choice; otherwise, you stand pat. What are your chances of winning the car?

Short answer

(a) $\frac{1}{3}$; (b) $\frac{2}{3}$; (c) $\frac{1}{2}$.

Step-by-step solution

Understanding the Problem

In the game, you want to win the car which is hidden behind one of the three doors. After you choose a door, the moderator opens one of the other two doors revealing a goat. You must then decide to stick with your original choice or switch to the other unopened door to maximize your chances of winning the car.

Evaluate Standing Pat

If you decide to stay with your original choice, you have a 1 in 3 chance of having picked the car initially since there are three doors and only one car. Therefore, the probability of winning by sticking to your original choice is $\frac{1}{3}$.

Evaluate Switching

By the process of elimination, if you picked a door with a goat initially (which has a probability of $\frac{2}{3}$), switching will win you the car, as the remaining unchosen door must have the car. Therefore, the probability of winning by switching is $\frac{2}{3}$.

Evaluate Coin Flip Decision

If you flip a coin to decide whether to switch or not, there is a 50% chance of switching and a 50% chance of not switching. Calculating the expected probability: $E=\frac{1}{2}\times \frac{1}{3}+\frac{1}{2}\times \frac{2}{3}=\frac{1}{6}+\frac{1}{3}=\frac{1}{2}$. Therefore, your probability of winning the car is $\frac{1}{2}$.

Key concepts

Game Theory

Game theory is a fundamental concept in probability and economics. It studies interactions where the outcome for each participant depends on the actions of others. In simple terms, it's about strategic decision-making. Whenever you're in a situation with other players involved, and your reward or punishment depends on your action and theirs, game theory is at play.
In the Monty Hall Problem, the contestant and the moderator represent two players. The contestant's goal is to win the sports car by choosing the right door. The moderator reveals a goat behind another door after the contestant’s initial choice. This action gives the game its strategic twist.

• Players: Contestant and moderator.
• Strategies: Choosing a door, deciding to switch or not.
• Outcomes: Winning a car or ending up with a goat.

The essence of game theory here is understanding the impact of the moderator's action, which changes the probabilities and suggests that a different strategy yields better outcomes. Understanding this strategy is the key to making better decisions under uncertainty.

Monty Hall Problem

The Monty Hall Problem is a classic example of probability puzzles. Named after the game show host Monty Hall, this problem challenges our intuition. Here's how it works:
Imagine you are on a game show. Three doors await you. Behind one door is a shiny sports car (which you want), and behind the other two are goats (less appealing!). Your goal is to pick the door with the car.

After your initial choice, Monty — knowing what's behind the doors — opens one of the other two doors, always revealing a goat. Now you face a crucial question: Do you stick with your first choice or switch to the other unopened door?

• If you stick with your first choice, you have a 1/3 chance of winning since you made your selection randomly from three doors.
• If you switch, you leverage the fact that Monty's action (revealing a goat) gives you a higher chance of winning. The probability of winning by switching is 2/3.

Most people's intuition is to think sticking or switching has the same odds, but the maths show otherwise — switching is a better strategy!

Decision Making in Uncertainty

Everyday decisions often involve uncertainty where outcomes aren’t guaranteed. The Monty Hall Problem provides a perfect example of this concept. How you make choices when the results aren't certain can significantly impact the outcome.
When faced with uncertainty in the Monty Hall Problem, your decision about whether to switch doors or not is crucial. Initially, all doors seem equally likely to hide the car. However, once Monty reveals a goat, the dynamics change.

• Standing pat (not switching): You have the same initial odds of 1/3, because nothing changes a selected door's chance of having the car.
• Switching: Choose to switch knowing that a new decision reassigns probabilities; here, it's 2/3 in favor of the car.
• Flipping a coin to decide: This method yields a 1/2 chance, essentially averaging out the probabilities of standing (1/3) and switching (2/3).

In uncertain situations, information and logic play an essential role. By understanding probability, you can navigate these scenarios more effectively, increasing your chances of a successful outcome.