Events A and B satisfy P(A) = 0.3 and P(A ∪ B) = 0.8. Assuming independence, find P(B).

Introduction / Context:
This problem combines the union formula with independence. When A and B are independent, P(A ∩ B) = P(A)P(B), which plugs into the union identity P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Given Data / Assumptions:

• P(A) = 0.3.
• P(A ∪ B) = 0.8.
• A and B are independent.

Concept / Approach:
Let p = P(B). Then 0.8 = 0.3 + p − 0.3p = 0.3 + 0.7p, so solve for p.

Step-by-Step Solution:
0.8 = 0.3 + 0.7p ⇒ 0.5 = 0.7p ⇒ p = 5/7 ≈ 0.7142857.

Verification / Alternative check:
Compute P(A ∩ B) = 0.3 * (5/7) = 1.5/7. Then P(A) + P(B) − P(A ∩ B) = 0.3 + 5/7 − 1.5/7 = 0.3 + 3.5/7 = 0.3 + 0.5 = 0.8, consistent.

Why Other Options Are Wrong:
3/8, 2/7, 4/7 do not satisfy 0.8 = 0.3 + p − 0.3p under independence.

Common Pitfalls:
Using the addition rule for mutually exclusive events (A ∪ B = A + B) instead of the correct union formula with intersection.

Final Answer:
5/7