Understanding the Problem: Marginal Probability Calculation

The question asks for the marginal probability of the variable 'Cavity' being true, denoted as \( P(\text{cavity}) \). We are given a full joint probability distribution table for three Boolean variables: Toothache, Cavity, and Catch.

A full joint probability distribution table lists the probability of every possible combination of values for all the variables in the domain. In this case, with three Boolean variables, there are \( 2^3 = 8 \) possible combinations, and their probabilities are listed in the table.

Analyzing the Full Joint Probability Distribution Table

The provided table is structured to show the joint probabilities for different states of Toothache, Cavity, and Catch. The rows represent the state of 'Cavity' (true or false), and the columns represent combinations of 'Toothache' and 'Catch'.

Each cell in the table gives the joint probability \( P(\text{Toothache}, \text{Catch}, \text{Cavity}) \). For example:

• \( P(\text{toothache}, \text{catch}, \text{cavity}) = 0.108 \)
• \( P(\text{toothache}, \text{￢catch}, \text{cavity}) = 0.012 \)
• \( P(\text{￢toothache}, \text{catch}, \text{cavity}) = 0.072 \)
• \( P(\text{￢toothache}, \text{￢catch}, \text{cavity}) = 0.008 \)
• \( P(\text{toothache}, \text{catch}, \text{￢cavity}) = 0.016 \)
• \( P(\text{toothache}, \text{￢catch}, \text{￢cavity}) = 0.064 \)
• \( P(\text{￢toothache}, \text{catch}, \text{￢cavity}) = 0.144 \)
• \( P(\text{￢toothache}, \text{￢catch}, \text{￢cavity}) = 0.576 \)

Calculating Marginal Probability P(cavity)

The marginal probability of a variable is obtained by summing the joint probabilities over all the possible states of the other variables. This process is called marginalization or summing out.

To find \( P(\text{cavity}) \), we need to sum the probabilities of all entries in the joint distribution where 'Cavity' is true, regardless of whether 'Toothache' is true or false, or whether 'Catch' is true or false.

From the table, the probabilities where 'Cavity' is true are found in the first row:

• \( P(\text{toothache}, \text{catch}, \text{cavity}) = 0.108 \)
• \( P(\text{toothache}, \text{￢catch}, \text{cavity}) = 0.012 \)
• \( P(\text{￢toothache}, \text{catch}, \text{cavity}) = 0.072 \)
• \( P(\text{￢toothache}, \text{￢catch}, \text{cavity}) = 0.008 \)

The marginal probability \( P(\text{cavity}) \) is the sum of these probabilities:

\( P(\text{cavity}) = P(\text{toothache}, \text{catch}, \text{cavity}) + P(\text{toothache}, \text{￢catch}, \text{cavity}) + P(\text{￢toothache}, \text{catch}, \text{cavity}) + P(\text{￢toothache}, \text{￢catch}, \text{cavity}) \)

\( P(\text{cavity}) = 0.108 + 0.012 + 0.072 + 0.008 \)

Step-by-Step Calculation

Let's perform the addition:

Step 1: Add the first two probabilities:

\( 0.108 + 0.012 = 0.120 \)

Step 2: Add the last two probabilities:

\( 0.072 + 0.008 = 0.080 \)

Step 3: Add the results from Step 1 and Step 2:

\( 0.120 + 0.080 = 0.200 \)

So, the marginal probability \( P(\text{cavity}) = 0.200 \).

Result

The marginal probability of Cavity being true, \( P(\text{cavity}) \), is 0.200.

Revision Table: Key Probability Concepts

Additional Information: Marginalization Process

Marginalization is a fundamental operation in probability theory, especially when working with joint distributions. It allows us to determine the probability distribution of a subset of variables without considering the values of the remaining variables. This is useful because often we are interested in the probability of a single event or a smaller set of events, rather than the probability of the full configuration of all variables.

For a set of variables \( X_1, X_2, \dots, X_n \) with a full joint distribution \( P(X_1, \dots, X_n) \), the marginal probability of a subset of these variables, say \( P(X_i) \), is calculated by summing the joint probability over all possible values of the variables other than \( X_i \). Mathematically, this is represented as:

\( P(X_i) = \sum_{x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n} P(X_1=x_1, \dots, X_i, \dots, X_n=x_n) \)

In our case, for \( P(\text{cavity}) \), the summation is over all possible states of 'Toothache' (toothache, ￢toothache) and 'Catch' (catch, ￢catch):

\( P(\text{cavity}) = \sum_{\text{Toothache} \in \{\text{toothache}, \text{￢toothache}\}} \sum_{\text{Catch} \in \{\text{catch}, \text{￢catch}\}} P(\text{Toothache}, \text{Catch}, \text{cavity}) \)

This confirms the method used to sum the four values from the table's 'cavity' row.