Case Study: The Monty Hall Problem — When Intuition Betrays You

The Setup

It's 1990, and Marilyn vos Savant — listed in the Guinness Book of World Records for the highest recorded IQ — writes a column for Parade magazine. A reader sends her a puzzle based on the game show Let's Make a Deal, hosted by Monty Hall:

You're on a game show. There are three doors. Behind one door is a car. Behind the other two are goats. You pick a door — say, Door 1. The host, who knows what's behind the doors, opens another door — say, Door 3 — revealing a goat. The host then asks: "Do you want to switch to Door 2?"

Should you switch?

Marilyn answered: Yes, you should switch. Switching gives you a 2/3 probability of winning the car.

The response was immediate and furious. Roughly 10,000 readers wrote in to tell her she was wrong — including nearly 1,000 people with PhDs. A math professor wrote: "You blew it, and you blew it big!" Another wrote: "Maybe women look at math problems differently than men." (They didn't look at the math very carefully before writing that one.)

But Marilyn was right. And the Monty Hall problem has become one of the most famous examples of how badly human intuition fails at probability.

Why People Get It Wrong

Most people's first instinct is: "There are two doors left. The car is behind one of them. So it's 50/50. Switching doesn't matter."

This feels completely logical. And it's completely wrong.

The error is subtle: it treats the host opening a door as if it provides no information. But the host's action is not random. Monty knows where the car is, and he always opens a door with a goat. That constraint changes the probabilities.

Let's work through it carefully.

The Probability Calculation

Your Initial Choice

When you pick Door 1, here are the probabilities:

You had a 1/3 chance of picking the car. That means there's a 2/3 chance the car is behind one of the other two doors.

After Monty Opens a Door

Now Monty opens a door (say Door 3) and shows you a goat. Let's see what happens in each scenario:

Scenario 1: The car is behind Door 1 (your door). Probability: 1/3. Monty can open either Door 2 or Door 3 (both have goats). He opens Door 3. If you switch → you get a goat. If you stay → you get the car.

Scenario 2: The car is behind Door 2. Probability: 1/3. Monty must open Door 3 (the only door with a goat that isn't yours). He opens Door 3. If you switch → you get the car. If you stay → you get a goat.

Scenario 3: The car is behind Door 3. Probability: 1/3. Monty can't open Door 3 (it has the car!). He opens Door 2 instead. This scenario is eliminated because we observed Monty opening Door 3.

Wait — let's reconsider. If Monty opened Door 3, then Scenario 3 either didn't happen or Monty would have opened a different door. In the two remaining scenarios where Monty opens Door 3:

Both scenarios were equally likely before you chose. After Monty opens Door 3:

• Staying wins in 1 out of 2 remaining scenarios → but that scenario had probability 1/3.
• Switching wins in 1 out of 2 remaining scenarios → and that scenario also had probability 1/3.

Actually, let's be more precise. Let's use the original probabilities directly:

• Probability the car is behind Door 1 (your pick): 1/3
• Probability the car is behind Door 2 (the remaining door): 2/3

Why 2/3? Because the 2/3 probability that the car was behind "one of the other two doors" hasn't changed. Monty simply concentrated that entire 2/3 onto the one remaining door by eliminating a goat door.

Result: Switching gives you a 2/3 chance of winning. Staying gives you 1/3.

Still Not Convinced? Let's Simulate It

This is a case where simulation can settle the argument once and for all.

import numpy as np

np.random.seed(42)

def monty_hall_simulation(n_games=100000, switch=True):
    """Simulate the Monty Hall problem."""
    wins = 0

    for _ in range(n_games):
        # Randomly place the car behind one of 3 doors (0, 1, 2)
        car_door = np.random.randint(0, 3)

        # Contestant picks a door (always door 0 for simplicity)
        contestant_pick = 0

        # Monty opens a door that:
        #   - is NOT the contestant's pick
        #   - does NOT have the car
        available_doors = [d for d in range(3)
                          if d != contestant_pick and d != car_door]
        monty_opens = available_doors[np.random.randint(0, len(available_doors))]

        if switch:
            # Switch to the remaining door
            remaining = [d for d in range(3)
                        if d != contestant_pick and d != monty_opens]
            final_pick = remaining[0]
        else:
            # Stay with original pick
            final_pick = contestant_pick

        if final_pick == car_door:
            wins += 1

    return wins / n_games

n_games = 100000

# Run both strategies
win_rate_switch = monty_hall_simulation(n_games, switch=True)
win_rate_stay = monty_hall_simulation(n_games, switch=False)

print(f"Results over {n_games:,} games:")
print(f"  Strategy: SWITCH → Win rate: {win_rate_switch:.4f} "
      f"(theoretical: 0.6667)")
print(f"  Strategy: STAY   → Win rate: {win_rate_stay:.4f} "
      f"(theoretical: 0.3333)")
print(f"\nSwitching wins approximately {win_rate_switch/win_rate_stay:.1f}x "
      f"as often as staying.")

Run this code yourself. Over 100,000 games, switching wins approximately 66.7% of the time. Staying wins approximately 33.3%. Marilyn was right.

Why This Problem Matters

The Monty Hall problem isn't just a party trick. It illustrates several profound lessons about probability:

1. Intuition Is Unreliable

Nearly everyone gets this wrong on first attempt — including mathematicians and statisticians. When approximately 1,000 PhD holders write to a magazine to say the correct answer is wrong, that tells you something important: your gut feeling about probability is not to be trusted. Formal calculation beats intuition every time.

This is exactly why the birthday problem from Section 8.1 is so surprising. Our brains aren't wired for probabilistic reasoning. We evolved to handle concrete, immediate threats — not abstract long-run frequencies. The rules in this chapter exist precisely because we need them. Our intuition alone isn't enough.

2. Information Changes Probabilities

The key to the Monty Hall problem is that Monty's action provides information. He didn't open a random door — he specifically opened a door he knew had a goat. That constrained choice transfers probability from the eliminated door to the remaining door.

This principle — that new information updates probabilities — is the foundation of Bayes' theorem, which you'll learn in Chapter 9. Medical tests, spam filters, and AI decision systems all work on this principle.

3. The "Equally Likely" Trap

The most common error is assuming that two remaining options are always 50/50. Just because there are two doors left doesn't mean each has an equal probability of hiding the car. The classical approach (Section 8.2) requires that outcomes be equally likely, and Monty's deliberate action destroyed that symmetry.

Watch for this error in real life: - "Either the treatment works or it doesn't — so the probability is 50/50." (No!) - "Either the defendant is guilty or innocent — so it's a coin flip." (No!) - "Either it will rain or it won't." (No!)

Having two outcomes does NOT mean they're equally probable.

The 100-Door Version

Still skeptical? Let's scale it up.

Imagine there are 100 doors. One has a car, 99 have goats. You pick Door 1. Monty, who knows where the car is, opens 98 doors — all with goats — leaving only your Door 1 and Door 73.

Would you switch to Door 73?

Of course you would. Your initial pick had a 1/100 chance of being right. There's a 99/100 chance the car is behind one of the other 99 doors. Monty just showed you which 98 of those 99 doors DON'T have the car, concentrating all that 99/100 probability onto a single door.

The 3-door version works exactly the same way — it's just less obvious because 1/3 versus 2/3 doesn't feel as dramatic as 1/100 versus 99/100.

Connection to the Chapter

The Monty Hall problem uses every concept from this chapter:

• Sample space: Three possible car locations, each initially equally likely (classical probability).
• Complement rule: P(car behind another door) = 1 − P(car behind your door) = 1 − 1/3 = 2/3.
• The danger of intuition: Formal rules beat gut feelings.
• Simulation as verification: When the math seems unbelievable, simulation provides empirical confirmation.
• Information matters: This previews conditional probability (Chapter 9), where Monty's revealed information formally updates our probabilities.

Discussion Questions

1. Why do you think so many people — including experts — get the Monty Hall problem wrong? What specific aspect of the problem triggers the incorrect intuition?

2. The simulation consistently shows that switching wins about 2/3 of the time. If someone still doesn't believe the math, what does the simulation tell us about the relationship between theoretical probability and empirical evidence?

3. What would happen if Monty opened a door randomly (without knowing where the car is) and happened to reveal a goat? Would switching still be advantageous? Why or why not? (This is a subtle question — think carefully about what changes.)

4. Find a real-world situation where people might fall into the same trap as the Monty Hall problem — assuming two remaining options are equally likely when they're not.

The Takeaway

The Monty Hall problem is humbling. It shows that even simple probability setups can defeat experts' intuitions. But it's also empowering: once you have the formal tools — sample spaces, probability rules, and the ability to simulate — you can get the right answer even when your gut leads you astray.

That's the promise of this entire chapter: the rules of probability work even when your intuition doesn't.