Puzzle | Monty Hall problem (original) (raw)

Last Updated : 6 Jun, 2026

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 1 or stick to door 2?.

What do you decide to do to have better chances of winning a car?

Check if you were right - full answer with solution below.

**Solution: The main trick is that the host would open the door with a 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 a 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.

The probability of winning a car by switching is higher than not switching. It is advantageous to switch.