Question

Given: \( P(B) = 0.8 \) and \( P(A \cup B) = 0.8 \), find \( P(A' \cup B') \).

Hint:

The formula \( P(A' \cup B') = 1 - P(A \cap B) \) is useful for finding the probability of the complement of a union. Use the inclusion-exclusion principle to find the intersection probability when needed.

Answer 1

We are given: \[ P(B) = 0.8 \quad \text{and} \quad P(A \cup B) = 0.8 \] We need to find \( P(A' \cup B') \). Step 1: Use the complement rule.
Recall that: \[ P(A' \cup B') = 1 - P(A \cap B) \] We need to find \( P(A \cap B) \).
Step 2: Use the inclusion-exclusion principle.
From the inclusion-exclusion principle, we know: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the known values: \[ 0.8 = P(A) + 0.8 - P(A \cap B) \]
Step 3: Solve for \( P(A \cap B) \).
Rearrange the equation to solve for \( P(A \cap B) \): \[ P(A \cap B) = P(A) + 0.8 - 0.8 = P(A) \]
Step 4: Use the result in the complement rule.
Substitute \( P(A \cap B) = P(A) \) into the complement rule: \[ P(A' \cup B') = 1 - P(A) \] Thus, the final solution depends on \( P(A) \), which is not given directly. If \( P(A) = 0.8 \), then: \[ P(A' \cup B') = 1 - 0.8 = 0.2 \] Final Answer
Final Answer: If \( P(A) = 0.8 \), then \( P(A' \cup B') = 0.2 \).