The Monty Hall Problem is one of the most famous and counterintuitive puzzles in probability. Inspired by a real American TV game show called “Let’s Make a Deal”, it tests how humans perceive chance, randomness, and decision-making. At first glance, the puzzle seems like a simple guessing game. But when you understand how the host’s actions affect the odds, the solution completely flips your intuition. It’s one of those puzzles that seem wrong until math proves you right.
Let’s explore the setup, logic, and full step-by-step solution to the Monty Hall Problem.
The Monty Hall Problem Setup and Rules
Here’s how the puzzle works:
- There are three doors: behind one is a car, and behind the other two are goats.
- You pick one door, say door 2.
- The host, Monty Hall, knows what’s behind each door. He then opens one of the other two doors that definitely has a goat behind it, for example, door 3.
- Now, you’re left with two doors: the one you picked (door 2) and one unopened door (door 1).
The host then asks:
“Do you want to stay with door 2 or switch to door 1?”
The question is simple: Should you switch doors to increase your chances of winning the car?
How to Solve the Monty Hall Problem?
Most people believe the chances are now 50/50 since two doors remain. But that’s not true. The correct strategy, and the one that surprises everyone, is that you should always switch.
Let’s break this down step by step.
Step 1: Understanding the Initial Odds
When you first choose a door (say door 2):
- The chance that the car is behind your door = 1/3.
- The chance that the car is behind one of the other doors = 2/3.
That’s where the magic begins.
Step 2: The Host’s Action Changes the Game
Monty opens one of the remaining doors, but only one that has a goat. This action gives you new information.
- He never opens the door with the car.
- Therefore, his choice is not random; it’s based on knowledge of where the car is.
This means the 2/3 probability that the car was behind one of the other two doors now transfers entirely to the one remaining unopened door.
Step 3: Step-by-Step Example (Doors 1, 2, and 3)
Let’s assume the car’s position varies randomly behind the three doors.
1. Car behind door 1
- Player picks door 2.
- Host opens door 3 (goat).
- Switching to door 1 → wins the car.
2. Car behind door 2
- Player picks door 2.
- Host opens door 3 (goat).
- Switching to door 1 → loses (car is behind door 2).
3. Car behind door 3
- Player picks door 2.
- Host opens door 1 (goat).
- Switching to door 3 → wins the car.
Step 4: Summary of Outcomes
| Car Location | Player’s Initial Choice | Host Opens | Switching Wins? |
|---|---|---|---|
| Door 1 | Door 2 | Door 3 | Yes |
| Door 2 | Door 2 | Door 3 | No |
| Door 3 | Door 2 | Door 1 | Yes |
In 2 out of 3 cases, switching wins the car.
In 1 out of 3 cases, staying wins.
So, switching gives a 2/3 (66.7%) chance of winning, while sticking gives only 1/3 (33.3%).
Final Answer: Always Switch
Mathematically and logically, you should always switch. Why?
Because your first choice had a 1/3 chance of being right, and the host’s action effectively transfers the 2/3 probability to the remaining unopened door. By switching, you double your odds of winning from 33% to 67%.
Why the Monty Hall Problem Feels So Wrong?
The Monty Hall Problem is so famous because it tricks intuition. Our brains assume that two remaining doors mean equal chances, but that ignores how the host’s behaviour changes the situation. Monty Hall knows where the car is, and his choice of door to open adds information that changes probabilities. That’s why many people, even math teachers and PhDs, have argued about this puzzle for years before accepting the result.
Real-World Example
Imagine you’re playing a game show:
- Out of 3 boxes, one has ₹10,00,000 in cash, and the others are empty.
- You pick one.
- The host, knowing what’s inside, opens one empty box.
- Would you switch?
Of course, you’d switch every time, because the chance that you initially picked the right one is just 1/3.
The remaining box now has twice the odds of being the winner.
The Monty Hall Problem Explained in Probability Terms
Let’s express this in simple probability math:
| Action | Probability of Winning |
|---|---|
| Stay with first choice | 1/3 |
| Switch to other door | 2/3 |
So, the best possible choice is clear: Switch every time.
Why the Monty Hall Problem is Popular?
This problem is loved by:
- Game theory enthusiasts - it shows how conditional probability changes outcomes.
- Interviewers - to test logical and mathematical reasoning.
- Teachers and psychologists - as an example of human bias and intuition error.
It perfectly demonstrates how our intuition can fail in probabilistic situations.
Similar Logic Puzzles with Answers
Here are a few other classic logic puzzles that, like the Monty Hall Problem, challenge reasoning and probability understanding:
1. The 100 Prisoners and Hats Puzzle – Team Logic
Setup: 100 prisoners must guess their hat color using a shared logic strategy.
Answer: Using parity, 99 survive for sure - and sometimes all 100.
2. The Blue Eyes Puzzle – Deductive Reasoning
Setup: People on an island must determine their own eye color with no direct communication.
Answer: They all leave on the nth night once logic reveals the truth.
3. The River Crossing Puzzle – Sequential Strategy
Setup: A farmer must carry a wolf, goat, and cabbage across a river safely.
Answer: Goat first, then wolf, then cabbage, then goat again.
4. The Two Doors Puzzle – Truth vs Lies
Setup: Two guards guard two doors, one leads to freedom, one to death. One guard always lies.
Answer: Ask, “What would the other guard say?” Then choose the opposite door.
5. The 3 Prisoners Problem – Probability Paradox
Setup: One of three prisoners will survive; the guard reveals one doomed prisoner.
Answer: The chance of survival remains ⅓ - not ½.