Monty Hall problem

The Monty Hall problem is a surprising probability puzzle:

• There are 3 doors—two hide goats, and one hides a car.
• You pick one door (let’s call it door 2), hoping it has the car.
• The game show host, Monty Hall, then looks at the other two doors (1 and 3) and opens one that has a goat behind it (Say 3). (If both doors have goats, he chooses one at random.)

He then says to you, "Do you want to pick door 2 or stick to door 1. What do you decide to have better chances of winning a car?

Solution: The main trick is that the host would open the door with goat only so the chances of the other door having a car are higher. Hence you should always switch to improve your chances. Below is detailed solution.

Let’s solve the Monty Hall problem step by step, assuming the gates are numbered 1, 2, and 3:

Setup:

• Player’s choice: The player initially picks gate 2.

The car is equally likely to be behind any of the three gates initially. Let’s evaluate the three possible arrangements:

1. Car behind gate 1:

• Player picks gate 2 (initial choice).
• Host must open gate 3, showing a goat (since gate 1 has the car).
• Switching to gate 1 wins the car.

2. Car behind gate 2:

• Player picks gate 2 (initial choice).
• Host opens gate 3, showing a goat.
• Switching to gate 1 loses, as the car is behind gate 2.

3. Car behind gate 3:

• Player picks gate 2 (initial choice).
• Host cannot open gate 3 because it has the car. Instead, he opens gate 1, showing a goat.
• Switching to gate 3 wins the car.

Summary of outcomes:

• In 2 out of 3 scenarios, switching wins the car.
• In 1 out of 3 scenarios, staying with the initial choice wins.

As probability of winning a car by switching is higher than not switching. It is advantage to switch.