A bag contains 5 red, 7 blue, and 8 green marbles. If two marbles are drawn at random without replacement, what is the probability both are green?
["Understanding the Probability of Drawing Two Green Marbles Without Replacement", "When dealing with probability questions involving combinations and without replacement, careful calculation is key. One classic example involves determining the likelihood of drawing specific colored marbles from a bag. In this article, we explore a straightforward probability scenario: what is the chance of drawing two green marbles from a bag containing 5 red, 7 blue, and 8 green marbles when two marbles are drawn at random without replacement?", "---", "### The Probability Problem at a Glance", "We are given:\n- 5 red marbles\n- 7 blue marbles\n- 8 green marbles", "Total marbles = 5 + 7 + 8 = 20 marbles", "We want to find the probability that both marbles drawn without replacement are green.", "---", "### Step-by-Step Calculation", "#### Step 1: Probability the First Marble Is Green", "There are 8 green marbles out of 20 total:", "[\nP(\ ext{First green}) = \frac{8}{20}\n]", "#### Step 2: Probability the Second Marble Is Green (Given the First Was Green)", "Since one green marble has already been removed, only 7 green marbles remain out of 19 total marbles:", "[\nP(\ ext{Second green} \mid \ ext{First green}) = \frac{7}{19}\n]", "#### Step 3: Multiply Probabilities for Both Events", "Since both draws depend on each other (without replacement), multiply the probabilities:", "[\nP(\ ext{Both green}) = \frac{8}{20} \ imes \frac{7}{19} = \frac{56}{380}\n]", "#### Step 4: Simplify the Fraction", "[\n\frac{56}{380} = \frac{14}{95} \quad \ ext{(dividing numerator and denominator by 4)}\n]", "---", "### Final Answer", "The probability that both marbles drawn at random without replacement are green is 14/95, which is approximately 0.1474 or 14.74%.", "---", "### Why This Probability Matters", "Understanding such probabilities helps in decision-making across many real-world scenarios—from quality control and games of chance to scientific studies and risk assessment. Mastering conditional probability ensures clearer insights into random events.", "---", "### Key Takeaways", "- Total marbles: 20\n- Green marbles: 8\n- Probability first marble is green: 8/20\n- Probability second green, given first was green: 7/19\n- Final probability: 14/95", "---", "Search Terms: probability of drawing two green marbles, conditional probability marble draw, math probability without replacement, marbles probability problem explanation", "Optimize this SEO-friendly article by integrating these keywords naturally, writing engaging introductory and concluding paragraphs, using FAQ-style subheadings for readability, and adding meta descriptions to boost visibility."]
