Calculating P(cavity V toothache) from Joint Probability Table

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

The full joint distribution table is provided as follows:

To find $P(A \lor B)$, we can use the principle of inclusion-exclusion:

$$P(A \lor B) = P(A) + P(B) - P(A \land B)$$

In this case, $A$ is 'cavity' and $B$ is 'toothache'. So we need to find $P(\text{cavity})$, $P(\text{toothache})$, and $P(\text{cavity} \land \text{toothache})$ from the joint distribution table.

Finding Marginal and Joint Probabilities

• $P(\text{cavity})$: This is the sum of all entries in the 'Cavity' row.

  $$P(\text{cavity}) = P(\text{cavity} \land \text{toothache} \land \text{catch}) + P(\text{cavity} \land \text{toothache} \land \neg\text{catch}) + P(\text{cavity} \land \neg\text{toothache} \land \text{catch}) + P(\text{cavity} \land \neg\text{toothache} \land \neg\text{catch})$$

  $$P(\text{cavity}) = 0.108 + 0.012 + 0.072 + 0.008 = 0.200$$

• $P(\text{toothache})$: This is the sum of all entries in the 'Toothache' columns (the first two columns).

  $$P(\text{toothache}) = P(\text{cavity} \land \text{toothache} \land \text{catch}) + P(\text{cavity} \land \text{toothache} \land \neg\text{catch}) + P(\neg\text{cavity} \land \text{toothache} \land \text{catch}) + P(\neg\text{cavity} \land \text{toothache} \land \neg\text{catch})$$

  $$P(\text{toothache}) = 0.108 + 0.012 + 0.016 + 0.064 = 0.200$$

• $P(\text{cavity} \land \text{toothache})$: This is the sum of entries where both 'Cavity' is true (Cavity row) AND 'Toothache' is true (Toothache columns). These are the first two entries in the table.

  $$P(\text{cavity} \land \text{toothache}) = P(\text{cavity} \land \text{toothache} \land \text{catch}) + P(\text{cavity} \land \text{toothache} \land \neg\text{catch})$$

  $$P(\text{cavity} \land \text{toothache}) = 0.108 + 0.012 = 0.120$$

Applying the Formula

Now substitute these values into the inclusion-exclusion formula:

$$P(\text{cavity} \lor \text{toothache}) = P(\text{cavity}) + P(\text{toothache}) - P(\text{cavity} \land \text{toothache})$$

$$P(\text{cavity} \lor \text{toothache}) = 0.200 + 0.200 - 0.120$$

$$P(\text{cavity} \lor \text{toothache}) = 0.400 - 0.120$$

$$P(\text{cavity} \lor \text{toothache}) = 0.280$$

Alternative Method: Using the Complement

Alternatively, we can use the formula $P(A \lor B) = 1 - P(\neg A \land \neg B)$.

In this case, $P(\text{cavity} \lor \text{toothache}) = 1 - P(\neg \text{cavity} \land \neg \text{toothache})$.

$P(\neg \text{cavity} \land \neg \text{toothache})$ is the sum of entries where 'Cavity' is false ($\neg$Cavity row) AND 'Toothache' is false ($\neg$Toothache columns). These are the last two entries in the table:

$$P(\neg \text{cavity} \land \neg \text{toothache}) = P(\neg\text{cavity} \land \neg\text{toothache} \land \text{catch}) + P(\neg\text{cavity} \land \neg\text{toothache} \land \neg\text{catch})$$

$$P(\neg \text{cavity} \land \neg \text{toothache}) = 0.144 + 0.576 = 0.720$$

Now apply the complement formula:

$$P(\text{cavity} \lor \text{toothache}) = 1 - P(\neg \text{cavity} \land \neg \text{toothache})$$

$$P(\text{cavity} \lor \text{toothache}) = 1 - 0.720$$

$$P(\text{cavity} \lor \text{toothache}) = 0.280$$

Both methods yield the same result.

Revision Table: Key Probabilities

Additional Information: Joint Distribution and Marginalization

A joint probability distribution table lists the probability of every possible combination of states for a set of random variables. For $n$ Boolean variables, there are $2^n$ possible combinations, and the sum of all probabilities in the table must equal 1.

To find the probability of a single variable or a subset of variables being in a specific state (this is called a marginal probability), you sum the probabilities of all the joint states that are consistent with that specific state. For example, to find $P(\text{cavity})$, we summed up all entries in the table where Cavity is true, regardless of the states of Toothache and Catch. This process is called marginalization.

The joint distribution table contains all the information needed to answer any probability query about the variables in the domain.