Consider a domain consisting of three Boolean variables Toothache, Cavity, and…
2021
Consider a domain consisting of three Boolean variables Toothache, Cavity, and Catch. The full joint distribution is a 2×2×2 table as shown in the figure below.
toothache | ¬toothache | |||
catch | ¬catch | catch | ¬catch | |
cavity | 0.108 | 0.012 | 0.072 | 0.008 |
¬cavity | 0.016 | 0.064 | 0.144 | 0.576 |
P(cavity ∨ toothache) is __________.
- A.
0.200
- B.
0.120
- C.
0.280
- D.
0.600
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Correct answer: C
Method 1: Inclusion-Exclusion
Using the formula P(A∨B)=P(A)+P(B)−P(A∧B):
P(cavity): Sum the top row (0.108+0.012+0.072+0.008)=0.200.
P(toothache): Sum the "toothache" column (0.108+0.012+0.016+0.064)=0.200.
P(cavity∧toothache): Sum where both are true (0.108+0.012)=0.120.
Result: 0.200+0.200−0.120= 0.280.
Method 2: Complement Rule
Using the formula 1−P(¬cavity∧¬toothache):
Identify the entries where neither cavity nor toothache is present. This is the bottom-right section under "¬toothache" and "¬cavity".
P(¬cavity∧¬toothache): Sum (0.144+0.576)=0.720.
Result: 1−0.720= 0.280.
Note: The joint probability P(cavity ∧ toothache) may be given directly or computed from a conditional probability (for example, P(toothache | cavity) × P(cavity)) when that conditional is available.