Problem 4. Random events A, B ⊆ Ω satisfy P(A ∪ B′ ) = 0.23, P(A′ ∪ B′ ) = 0.81.

Calculate P(B).

Prove that if P(A) < 0.21, then P(A′ ∩ B′) > 0.02.

To solve this, let's analyze the given probabilities and express them in terms of probabilities of \( A \) and \( B \) and their complements.

### Step 1: Analyze Given Probabilities
1. **Given**: \( P(A \cup B') = 0.23 \)
   - Using the formula for the probability of a union:
     \[
     P(A \cup B') = P(A) + P(B') - P(A \cap B')
     \]

2. **Given**: \( P(A' \cup B') = 0.81 \)
   - Similarly, we apply the formula for a union here:
     \[
     P(A' \cup B') = P(A') + P(B') - P(A' \cap B')
     \]
   - Since \( P(A') = 1 - P(A) \) and \( P(B') = 1 - P(B) \), we can rewrite this as:
     \[
     P(A' \cup B') = (1 - P(A)) + (1 - P(B)) - P(A' \cap B')
     \]
   - Simplifying further:
     \[
     0.81 = 1 - P(A) - P(B) + P(A \cap B)
     \]

### Step 2: Express \( P(B) \) in Terms of \( P(A) \)
From the two given equations, we can isolate and solve for \( P(B) \) by solving a system of equations using \( P(A) \

Problem 4. Random events A, B ⊆ Ω satisfy P(A ∪ B′ ) = 0.23, P(A′ ∪ B′ ) = 0.81. Calculate P(B). Prove that if P(A) < 0.21, then P(A′ ∩ B′) > 0.02.

Explore how to calculate P(B) when given the probabilities P(A ∪ B′) = 0.23 and P(A′ ∪ B′) = 0.81, and prove that if P(A) < 0.21, then P(A′ ∩ B′) > 0.02. This problem covers key concepts in probability theory, including set operations and complementary events. Suitable for college-level students or advanced high school learners.

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