Question:** A box contains 3 red, 4 blue, and 5 green balls. If two balls are drawn at random, what is the probability that both are red?
["Probability of Drawing Two Red Balls: A Complete Guide", "When it comes to probability problems involving balls in a box, one of the most common and insightful questions is: What is the probability that both balls drawn at random are red? This scenario neatly combines basic probability principles with practical counting techniques, making it an excellent example for learners and math enthusiasts alike.", "In this article, we’ll break down the problem step-by-step, explain how to calculate the probability, and highlight why understanding combinations is key to solving such questions.", "---", "### Problem Setup", "We have a box containing:", "- 3 red balls\n- 4 blue balls\n- 5 green balls", "Total number of balls:\n[\n3 + 4 + 5 = 12\n]", "We draw two balls at random without replacement, and we want the probability that both balls are red.", "---", "### Step 1: Total Number of Possible Outcomes", "Drawing two balls from 12 isn’t just one outcome—it’s all possible unique pairs. Since order doesn’t matter, we use combinations:", "[\n\ ext{Total ways to choose 2 balls from 12} = \binom{12}{2} = \frac{12 \ imes 11}{2} = 66\n]", "---", "### Step 2: Number of Favorable Outcomes", "We want both balls to be red. There are 3 red balls, and we want to choose 2 of them:", "[\n\ ext{Ways to choose 2 red balls from 3} = \binom{3}{2} = \frac{3 \ imes 2}{2} = 3\n]", "---", "### Step 3: Calculating the Probability", "Probability is the ratio of favorable outcomes to total possible outcomes:", "[\nP(\ ext{both red}) = \frac{\ ext{Favorable outcomes}}{\ ext{Total outcomes}} = \frac{\binom{3}{2}}{\binom{12}{2}} = \frac{3}{66} = \frac{1}{22}\n]", "So, the probability of drawing two red balls is:", "[\n\boxed{\frac{1}{22}}\n]", "---", "### Why Use Combinations?", "Using combinations instead of permutations ensures we count each unique pair only once and avoid overcounting. Since drawing red-red is an unordered event, it fits naturally in the combination model.", "---", "### Practical Tips for Similar Problems", "- Always calculate total outcomes using combinations when order doesn’t matter.\n- Target only the combinations relevant to your condition (e.g., all red pairs).\n- Simplify fractions early to keep your result clean and understandable.", "---", "### Conclusion", "Understanding how to compute probabilities in such scenarios builds a solid foundation in probability theory. The question “what is the probability both drawn balls are red?” is not just a math drill—it’s a gateway to reasoning about risk, selection, and uncertainty in real life.", "Next time you see a similar question, break it down using combinations, calculate carefully, and enjoy the clarity probability brings to randomness!", "---", "### Related Keywords for SEO:\nProbability of two red balls, probability question with balls, two balls drawn probability, probability of drawing red balls, combinatorics probability example, drawing balls probability guide", "---", "Understanding probability is easier with real problems—dive in and try your own exercises!"]
