A box contains 5 red, 4 blue, and 6 green marbles. If two marbles are drawn at random without replacement, what is the probability that both are green?
["Title: Probability of Drawing Two Green Marbles: A Step-by-Step Guide", "Meta Description: Learn how to calculate the probability of drawing two green marbles from a box containing 5 red, 4 blue, and 6 green marbles. Find the exact probability without replacement.", "---", "Understanding the Probability of Drawing Two Green Marbles", "Probability is a fundamental concept in statistics that helps us predict outcomes in uncertain situations. A classic example involves drawing marbles from a container—a clear and intuitive way to explore probability without complex formulas.", "In this article, we explore a scenario involving colored marbles: a box containing 5 red, 4 blue, and 6 green marbles, totaling 15 marbles. We’ll calculate the probability that both marbles drawn at random without replacement are green.", "---", "### The Marble Composition", "Let’s summarize the number of marbles:", "- Red: 5\n- Blue: 4\n- Green: 6\n- Total: 5 + 4 + 6 = 15 marbles", "---", "### Steps to Calculate the Probability", "To find the probability that both marbles drawn are green, follow these steps:", "Step 1: Find the probability of drawing a green marble first", "There are 6 green marbles out of 15 total marbles:", "[\nP(\ ext{First green}) = \frac{6}{15}\n]", "Step 2: Determine the probability of drawing a second green marble", "After drawing one green marble without replacement:", "- Number of green marbles left: 6 − 1 = 5\n- Total marbles remaining: 15 − 1 = 14", "So,", "[\nP(\ ext{Second green} \mid \ ext{First green}) = \frac{5}{14}\n]", "Step 3: Multiply the probabilities", "Since the draws are dependent (without replacement), multiply the two probabilities:", "[\nP(\ ext{Both green}) = \frac{6}{15} \ imes \frac{5}{14}\n]", "Simplify the expression step-by-step:", "[\n= \frac{6 \ imes 5}{15 \ imes 14} = \frac{30}{210} = \frac{1}{7}\n]", "---", "### Final Answer", "The probability that both marbles drawn at random without replacement are green is:", "[\n\boxed{\frac{1}{7}}\n]", "---", "### Why This Matters", "This simple probability problem demonstrates how dependence between events affects outcomes. It’s widely used in engineering, finance, and data science to model real-world scenarios like quality control, survey sampling, and risk analysis.", "By breaking down the problem into its component probabilities, you build a strong foundation for tackling more complex problems involving combinations, conditional probability, and random sampling.", "---", "Keywords: probability, green marbles, draw two marbles, without replacement, probability calculation, combinatorics, probability examples, chance, statistics tutorial", "Related Searches:\n- Probability both marbles are green\n- Drawing marbles from a box\n- Conditional probability marble draw\n- Probability without replacement marbles\n- Green marble probability problem", "---", "### Summary", "- Total marbles: 15\n- Green marbles: 6\n- Probability first green: 6/15\n- Probability second green (after one green drawn): 5/14\n- Joint probability: (6/15) × (5/14) = 1/7", "Next time you’re faced with a marble or card draw problem, remember this step-by-step approach—simple math makes complex probabilities clear!"]
