Probability often becomes clearer when we look at real problems. One of the most famous examples used in statistics courses is the Monty Hall Problem, which demonstrates how probabilities change when new information appears.

In this learning post, we focus on a small but very important step in the solution: why we expand the probability (P(D_2)) using the Law of Total Probability.

The Setup of the Monty Hall Problem

Imagine a game show with three doors:

Behind one door is a car (the prize).
Behind the other two doors are goats.

You choose Door 1.

The host, Monty, knows where the car is. After you choose Door 1, Monty opens another door that contains a goat.

In our situation:

Monty opens Door 2.

Now only two unopened doors remain:

Door 1 (your original choice)
Door 3 (the remaining closed door)

You are given the option to switch doors.

The key question becomes:

What is the probability that switching will win the car, given that Monty opened Door 2?

Mathematically we write this as:

$P(C_3 \mid D_2)$

which means:

“What is the probability that the car is behind Door 3, given that Monty opened Door 2?”

Step 1: Why Bayes’ Rule Is Needed

Initially, before Monty opens any door, the car could be behind any door with equal probability:

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

But once Monty opens Door 2, we receive new information.

This new information changes the probabilities.
To update probabilities after observing something, we use Bayes’ Rule.

Bayes’ Rule says:

$P(C_3 \mid D_2) = \frac{P(D_2 \mid C_3)P(C_3)}{P(D_2)}$

In plain English, this means:

Probability the car is behind Door 3 after seeing Door 2 opened

equals

(probability Monty would open Door 2 if the car were behind Door 3)

× (probability the car was behind Door 3 originally)

divided by

(probability that Monty opens Door 2 overall)

Step 2: Why We Must Expand (P(D_2))

The denominator asks:

What is the total probability that Monty opens Door 2?

But Monty’s behavior depends on where the car actually is.

There are three possible situations:

The car is behind Door 1
The car is behind Door 2
The car is behind Door 3

Monty might open Door 2 in some of these situations, but not in others.

So to compute the total probability that Monty opens Door 2, we must consider all possible locations of the car.

This is exactly what the Law of Total Probability does.

Step 3: Applying the Law of Total Probability

We calculate the probability that Monty opens Door 2 by adding the probabilities of all scenarios where this could happen.

$P(D_2) = P(D_2 \mid C_1)P(C_1) + P(D_2 \mid C_2)P(C_2) + P(D_2 \mid C_3)P(C_3)$

In plain English this means:

The probability that Monty opens Door 2 equals the sum of:

Probability the car is behind Door 1 and Monty opens Door 2
Probability the car is behind Door 2 and Monty opens Door 2
Probability the car is behind Door 3 and Monty opens Door 2

We add these possibilities together to get the total probability that Door 2 is opened.

Key Intuition

Think of the denominator as answering this question:

“Across all possible worlds where the car might be, how often would Monty open Door 2?”

Once we know that overall probability, Bayes’ Rule allows us to update our belief about where the car actually is.

Takeaway

Bayes’ Rule is simply a tool for updating probabilities after new information appears.

In the Monty Hall problem, the new information is:

Monty opened Door 2.

Using Bayes’ Rule together with the Law of Total Probability, we adjust our beliefs and discover why switching doors gives a higher chance of winning.

In the next step, we substitute the probabilities into the formula and obtain the result: