Question

For two events $A$ and $B$, $P(A \cup B) = \frac{5}{6}$, $P(A) = \frac{1}{6}$, $P(B) = \frac{2}{3}$, then $A$ and $B$ are

• A. independent
• B. mutually exhaustive
• C. mutually exclusive
• D. complementary

Hint:

Always glance at the values first! Notice that $P(A) + P(B) = \frac{1}{6} + \frac{4}{6} = \frac{5}{6}$. Because the individual probabilities sum precisely up to the union probability, it tells you immediately without rewriting the entire formula that there is zero overlap ($P(A \cap B) = 0$), identifying them as mutually exclusive right away.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem presents the standalone and union probabilities of two specific structural events, $A$ and $B$. We are tasked with determining the logical relationship between them from the provided classifications.

Step 2: Key Formula or Approach:
Use the Addition Theorem of Probability to calculate the probability of the simultaneous intersection of both events: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ Analyze the resulting value of $P(A \cap B)$: If $P(A \cap B) = 0$, the events are mutually exclusive. If $P(A \cap B) = P(A) \times P(B)$, the events are independent.

Step 3: Detailed Explanation:
Let's substitute the given values into the addition theorem formula: $$\frac{5}{6} = \frac{1}{6} + \frac{2}{3} - P(A \cap B)$$ Make the denominators uniform to combine fractions smoothly: $$\frac{2}{3} = \frac{4}{6}$$ Now substitute back: $$\frac{5}{6} = \frac{1}{6} + \frac{4}{6} - P(A \cap B)$$ $$\frac{5}{6} = \frac{5}{6} - P(A \cap B)$$ Isolating $P(A \cap B)$ yields: $$P(A \cap B) = \frac{5}{6} - \frac{5}{6} = 0$$ Since the intersection probability evaluates to exactly zero, it implies that events $A$ and $B$ cannot possibly occur at the same time. Therefore, they are mutually exclusive events.

Step 4: Final Answer:
Since $P(A \cap B) = 0$, the events are mutually exclusive, corresponding to option (C).