Question: A bag contains 5 red chips, 7 green chips, and 3 blue chips. If 4 chips are drawn without replacement, what is the probability that exactly 2 are green and 2 are red?

["Understanding the Probability of Drawing 2 Green and 2 Red Chips: A Detailed Statistics Breakdown", "When it comes to probability problems involving drawn items without replacement, clarity and precision are key. One commonly explored question involves a bag containing a mix of chips—red, green, and blue—and asks about the likelihood of drawing a specific combination. This article explores the probability of drawing exactly 2 green and 2 red chips from a bag containing 5 red, 7 green, and 3 blue chips when 4 chips are drawn without replacement.", "---", "### The Setup: Understanding the Composition of the Bag", "Before calculating the probability, let’s clearly define the scenario:", "- Red chips: 5\n- Green chips: 7\n- Blue chips: 3\n- Total chips: (5 + 7 + 3 = 15)", "We are drawing 4 chips without replacement, and we want the probability that exactly 2 are green and 2 are red.", "---", "### Why This Probability Matters", "This type of hypergeometric probability problem—where sampling is done without replacement and from a finite population—is essential in fields such as quality control, genetics, and lottery analysis. It enables us to predict outcomes in real-world situations involving fixed pool populations.", "---", "### Step-by-Step Probability Calculation", "We use the hypergeometric distribution for such sampling without replacement. The formula for the probability of drawing exactly (k_1) chips of type 1, (k_2) of type 2, and (k_3) of type 3 (with total draws (n)) is:", "[\nP(X_1 = k_1, X_2 = k_2, X_3 = k_3) = \frac{ \binom{R}{k_1} \binom{G}{k_2} \binom{B}{k_3} }{ \binom{T}{n} }\n]", "Where:\n- ( R ) = number of red chips = 5\n- ( G ) = number of green chips = 7\n- ( B ) = number of blue chips = 3\n- ( T ) = total chips = 15\n- ( n ) = chips drawn = 4\n- ( k_1 = 2 ) (green), ( k_2 = 2 ) (red), ( k_3 = 0 ) (blue) — because 2 + 2 = 4, and no blue chips are drawn", "---", "### Applying the Values", "First, calculate the number of favorable outcomes:", "- Choosing 2 green chips from 7:\n [\n \binom{7}{2} = \frac{7 \ imes 6}{2 \ imes 1} = 21\n ]", "- Choosing 2 red chips from 5:\n [\n \binom{5}{2} = \frac{5 \ imes 4}{2 \ imes 1} = 10\n ]", "- Choosing 0 blue chips from 3:\n [\n \binom{3}{0} = 1\n ]", "Multiply these together:\n[\n21 \ imes 10 \ imes 1 = 210\n]", "Now, calculate the total number of possible outcomes—any 4 chips from 15:\n[\n\binom{15}{4} = \frac{15 \ imes 14 \ imes 13 \ imes 12}{4 \ imes 3 \ imes 2 \ imes 1} = 1365\n]", "---", "### Final Probability", "The desired probability is:", "[\nP(2\ \ ext{green and }2\ \ ext{red}) = \frac{210}{1365}\n]", "Simplify the fraction by dividing numerator and denominator by their greatest common divisor, 105:", "[\n\frac{210 \div 105}{1365 \div 105} = \frac{2}{13}\n]", "So, the probability is exactly:", "[\n\frac{2}{13} \approx 0.1538 \quad \ ext{or} \quad 15.38%\n]", "---", "### Key Takeaways", "- This probability relies on hypergeometric principles due to finite population sampling without replacement.\n- The favorable outcomes involve combinations—not permutations—since the order of drawing doesn’t matter.\n- Simplifying fractions ensures clarity and a clean final answer useful in reporting or further analysis.", "---", "### Conclusion", "Understanding chip-drawing probabilities helps decode real-world probability concepts relevant in research, manufacturing, and decision-making. The chance of drawing exactly 2 green and 2 red chips from this bag is 2/13, a clean and precise result grounded in sound statistical principles. Whether for classrooms, games, or data analysis, mastering such problems builds a strong foundation in probability.", "---", "### Related Keywords for SEO Optimization", "- Probability of drawing two green and two red chips\n- Hypergeometric distribution example\n- Probability calculation without replacement\n- Chip draw probability problem\n- Combinatorics in chance problems\n- Red green blue chip probability sheet\n- Probability tutorial step-by-step", "If optimized, this SEO article ranks well for educational search queries, combining clear explanation with precise math, appealing to students, teachers, and statistics enthusiasts."]