This question asks us to calculate a conditional probability, specifically the probability of having a cavity given that the person has a toothache, using the provided full joint probability distribution table.

The full joint distribution table shows the probability for every possible combination of the three Boolean variables: Toothache, Cavity, and Catch.

Let's denote:

• T for Toothache
• C for Cavity
• A for Catch

The table can be interpreted as follows:

We need to find the conditional probability \(P(\text{Cavity} | \text{Toothache})\). The formula for conditional probability \(P(X | Y)\) is given by:

\[P(X | Y) = \frac{P(X \text{ and } Y)}{P(Y)}\]

In our case, X is 'Cavity' (meaning Cavity is true) and Y is 'Toothache' (meaning Toothache is true). So we need to calculate:

\[P(\text{Cavity} | \text{Toothache}) = \frac{P(\text{Cavity and Toothache})}{P(\text{Toothache})}\]

First, let's calculate \(P(\text{Cavity and Toothache})\). This is the sum of all probabilities where both Cavity and Toothache are true. Looking at the table, Cavity is true in the first row, and Toothache is true in the first two columns. The joint probabilities where both are true are \(P(C \text{ and } T \text{ and } A)\) and \(P(C \text{ and } T \text{ and } \neg A)\).

\[P(\text{Cavity and Toothache}) = P(C, T, A) + P(C, T, \neg A)\]

From the table:

\[P(\text{Cavity and Toothache}) = 0.108 + 0.012 = 0.120\]

Next, let's calculate \(P(\text{Toothache})\). This is the sum of all probabilities where Toothache is true, regardless of the state of Cavity or Catch. Looking at the table, Toothache is true in the first two columns. The joint probabilities where Toothache is true are \(P(T, C, A)\), \(P(T, C, \neg A)\), \(P(T, \neg C, A)\), and \(P(T, \neg C, \neg A)\).

\[P(\text{Toothache}) = P(T, C, A) + P(T, C, \neg A) + P(T, \neg C, A) + P(T, \neg C, \neg A)\]

From the table:

\[P(\text{Toothache}) = 0.108 + 0.012 + 0.016 + 0.064\]

\[P(\text{Toothache}) = (0.108 + 0.012) + (0.016 + 0.064) = 0.120 + 0.080 = 0.200\]

Now we can calculate the conditional probability \(P(\text{Cavity} | \text{Toothache})\) using the formula:

\[P(\text{Cavity} | \text{Toothache}) = \frac{P(\text{Cavity and Toothache})}{P(\text{Toothache})} = \frac{0.120}{0.200}\]

\[P(\text{Cavity} | \text{Toothache}) = \frac{0.120}{0.200} = \frac{120}{200} = \frac{12}{20} = \frac{3}{5} = 0.600\]

Therefore, the probability of a cavity, given evidence of a toothache, is 0.600.

Probability Calculation Revision Table

Additional Information on Probabilistic Inference

Probabilistic inference is the task of computing the probability of a query proposition given observed evidence. In this problem, the query is 'Cavity is true' and the evidence is 'Toothache is true'.

• Full Joint Distribution: A table listing the probability of every atomic event (every possible combination of variable values). It contains all information needed to answer any probabilistic query about the domain.
• Marginalization: To find the probability of a subset of variables (like \(P(\text{Toothache})\)), we sum the probabilities from the full joint distribution over all possible values of the other variables (Cavity and Catch in this case). This is also known as summing out.
• Conditional Probability: \(P(X | Y)\) represents the probability of event X occurring given that event Y has already occurred. It is calculated by dividing the joint probability of X and Y by the probability of Y.

Using the full joint distribution is a fundamental way to perform probabilistic inference, although it can be computationally expensive for domains with many variables.