The probability of event \( A \) is 0.4, and the probability of event \( B \) is 0.5. If \( A \) and \( B \) are independent, what is the probability of both \( A \) and \( B \) occurring?

["Understanding Independent Events: Calculating the Probability of Both Events Occurring", "In probability theory, one of the foundational concepts is the idea of independent events and how their probabilities combine. A classic question in introductory probability is determining the likelihood that two independent events both occur. This article explores a fundamental case: when the probability of event ( A ) is 0.4 and the probability of event ( B ) is 0.5, and ( A ) and ( B ) are independent. What is the chance that both events happen?", "### What Are Independent Events?", "Two events ( A ) and ( B ) are said to be independent if the occurrence of one does not affect the probability of the other. Mathematically, this means:", "[\nP(A \cap B) = P(A) \ imes P(B)\n]", "This product rule is the cornerstone for computing joint probabilities when independence holds.", "### Applying the Values", "Given:\n- ( P(A) = 0.4 )\n- ( P(B) = 0.5 )\n- Events ( A ) and ( B ) are independent", "Using the independence rule:", "[\nP(A \cap B) = P(A) \ imes P(B) = 0.4 \ imes 0.5 = 0.2\n]", "Thus, the probability that both ( A ) and ( B ) occur is 0.2, or 20%.", "### Why This Matters in Real-World Applications", "Understanding how to compute the joint probability of independent events is essential across many fields, including risk assessment, finance, medicine, and machine learning. For example, in quality control, estimating the likelihood of two independent defects occurring can inform better testing protocols. In predictive modeling, recognizing event independence simplifies complex probability calculations.", "### Summary", "For independent events ( A ) and ( B ):", "[\nP(A \ ext{ and } B) = P(A) \ imes P(B) = 0.4 \ imes 0.5 = 0.2\n]", "So, the probability that both events occur is 0.2 — a clear illustration of how independence allows straightforward multiplication of probabilities.", "---", "Keywords: probability of independent events, compute probability, P(A and B), independent events, probability calculation, conditional independence, mathematical expectation in probability, probability theory fundamentals."]