Question:** A bag contains 3 red, 4 blue, and 5 green marbles. Two are drawn without replacement. What is the probability both are green?
["Probability of Drawing Two Green Marbles – Step-by-Step Explained", "If a bag contains 3 red, 4 blue, and 5 green marbles, what is the chance—without replacement—that both marbles drawn are green? This classic probability question helps illustrate how to calculate probabilities in scenarios involving independent events with finite outcomes.", "### The Setup", "The bag holds a total of:", "- 3 red marbles\n- 4 blue marbles\n- 5 green marbles", "Total marbles = 3 + 4 + 5 = 12", "We are drawing two marbles without replacement, meaning the first marble is not returned before drawing the second. We want both marbles to be green.", "---", "### Step 1: Probability of first marble being green", "There are 5 green marbles out of 12 total:", "[\nP(\ ext{first is green}) = \frac{5}{12}\n]", "---", "### Step 2: Probability of second marble being green (given first was green)", "After removing one green marble, only 4 green marbles remain out of a total of 11 remaining marbles:", "[\nP(\ ext{second is green | first was green}) = \frac{4}{11}\n]", "---", "### Step 3: Multiply probabilities (since events are dependent)", "[\nP(\ ext{both green}) = P(\ ext{first green}) \ imes P(\ ext{second green} \mid \ ext{first green}) = \frac{5}{12} \ imes \frac{4}{11}\n]", "Calculate:", "[\n\frac{5}{12} \ imes \frac{4}{11} = \frac{20}{132} = \frac{5}{33}\n]", "---", "### Final Answer", "The probability that both marbles drawn without replacement are green is:", "[\n\boxed{\frac{5}{33} \approx 0.1515 \ ext{ or } 15.15%}\n]", "---", "### Why This Matters", "Understanding such probabilities is valuable in games, statistics, quality control, and decision-making under uncertainty. The key insight is recognizing that without replacement, the second event’s probability is conditional on the first—making both draws dependent processes.", "Whether you’re a student, teacher, or statistical enthusiast, grasping this concept strengthens your grasp of probability fundamentals.", "---", "Keywords: probability of drawing two green marbles, draw two marbles without replacement, 3 red 4 blue 5 green marbles, conditional probability, probability calculation, finite population, statistical probability, math tutorial."]
