Many people first encounter this puzzle from the TV show Let’s Make a Deal. But today we’ll ignore psychology and solve it purely using probability theory.

🚪 Problem Setup

3 doors
1 car
2 goats
You choose Door 1
The host opens a goat door
You switch

Question:

What is the probability of winning if you always switch?

📌 Step 1: Define Events

Let:

(C_j) = Car is behind Door (j)
(W) = We win using the switching strategy

Since the car is randomly placed:

$P(C_1) = P(C_2) = P(C_3) = \frac{1}{3}$

📌 Step 2: Apply the Law of Total Probability

The law states:

$P(W) = P(W \mid C_1)P(C_1) + P(W \mid C_2)P(C_2) + P(W \mid C_3)P(C_3)$

This means:

Overall probability = weighted average of conditional probabilities.

📌 Step 3: Compute Each Conditional Probability

Assume you initially picked Door 1.

Case 1: Car is behind Door 1

If you switch, you lose.

$P(W \mid C_1) = 0$

Case 2: Car is behind Door 2

If you switch, you win.

$P(W \mid C_2) = 1$

Case 3: Car is behind Door 3

If you switch, you win.

$P(W \mid C_3) = 1$

📌 Step 4: Substitute into Formula

$P(W) = 0 \cdot \frac{1}{3} + 1 \cdot \frac{1}{3} + 1 \cdot \frac{1}{3} = \frac{2}{3}$

🎯 Final Result

$P(W) = \frac{2}{3}$

💡 Why Did We Multiply by 0, 1, and 1?

Because:

If your initial choice was correct → switching guarantees loss → probability 0
If your initial choice was wrong → switching guarantees win → probability 1

Switching wins in 2 out of 3 possible worlds.

🔥 Key Takeaway for Students

When solving probability problems:

Define clear events
Break into cases
Use conditional probabilities
Apply the Law of Total Probability

No paradox.

No magic.

Just structured reasoning.

The Monty Hall Problem — Solved Using the Law of Total Probability

🚪 Problem Setup

📌 Step 1: Define Events

📌 Step 2: Apply the Law of Total Probability