\( P(A \cup B) = 0.4 + 0.5 - 0.2 = 0.7 \)

["# Understanding ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ): A Simple Guide to Probability of Combined Events", "Understanding probability is essential in fields like statistics, data science, and risk analysis. One fundamental formula in probability theory is the addition rule for complementary events:", "[\nP(A \cup B) = P(A) + P(B) - P(A \cap B)\n]", "This equation calculates the probability that either event ( A ) or event ( B ) occurs — or both. Let’s break down the example formula ( P(A \cup B) = 0.4 + 0.5 - 0.2 = 0.7 ) and explore what it means.", "## What Does ( P(A \cup B) ) Represent?", "( P(A \cup B) ) stands for the probability that event ( A ) happens, or event ( B ) happens, or both. This is useful when calculating the likelihood of any of these outcomes, avoiding double-counting overlaps.", "### Applying the Formula Step-by-Step", "Given:\n[\nP(A \cup B) = 0.4 + 0.5 - 0.2\n]\nThis means:", "- ( P(A) = 0.4 ) (probability of event ( A ))\n- ( P(B) = 0.5 ) (probability of event ( B ))\n- ( P(A \cap B) = 0.2 ) (probability both ( A ) and ( B ) occur)", "Plugging in:\n[\nP(A \cup B) = 0.4 + 0.5 - 0.2 = 0.7\n]", "### Interpretation\nThere’s a 70% chance that at least one of events ( A ) or ( B ) occurs—accounting for overlap so we don’t overestimate. Without subtracting ( P(A \cap B) ), we’d incorrectly double-count the 20% probability where both events happen.", "## Why This Formula Matters", "This equation is foundational because it helps compound probabilities in real-world scenarios:\n- Predicting customer behavior in marketing campaigns\n- Assessing risks in insurance or finance\n- Analyzing outcomes in scientific experiments", "理解 how to properly combine probabilities ensures accurate decision-making based on empirical data rather than flawed intuition.", "## Final Thoughts", "Remember:\n- ( P(A \cup B) ) = Probability of “A or B” including shared outcomes\n- Subtracting ( P(A \cap B) ) corrects for double-counting overlapping events", "The result ( 0.7 ) reflects a 70% chance that at least one of the events occurs—a clear win from using ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ).", "Whether you're studying statistics or applying probability in practice, mastering this formula sharpens your analytical rigor and improves probability-based predictions."]