A box contains 12 red balls, 8 blue balls, and 10 green balls. If three balls are drawn at random without replacement, what is the probability that all three balls are of different colors? - Decision Point

Understanding the Context

Across the United States, interest in interactive learning and hands-on data exploration continues to rise. Educational platforms, career development resources, and even casual social media conversations increasingly turn to structured probability challenges to build curiosity and sharpen analytical skills. In homes, classrooms, and personal study spaces, questions about color-based odds—like this one—resonate because they feel intuitive, tangible, and accessible. They spotlight how simple components combine in predictable yet surprising ways—ideal for mental engagement fueled by mobile-focused content strategies.

How It Actually Works: A Clear Breakdown

To calculate the probability of drawing one red, one blue, and one green ball in any order—without replacement—we begin by identifying the total number of balls: 12 red + 8 blue + 10 green = 30 balls total. The key is computing favorable outcomes over total possibilities.

Drawing balls without replacement means each selection changes the composition of the pool. We compute the probability across three sequential draws:

Key Insights

• First ball any color: 30 choices
• Second ball from a different color: after removing one, 29 left—so the number depends on which first color was picked
• Third ball of the remaining color: only one left for that hue

By summing across all valid sequences—adjusting for adjusting probabilities based on prior draws—we arrive at:

P(all different colors) = (12×8×10 × 6) ÷ [30 × 29 × 28]

This calculation accounts for all permutations of red-blue-green (3! = 6), weighted by likelihood, giving a precise probability grounded in combinatorial math. The result highlights how specific proportions—like the 12 red as the most common—shape outcomes without explicit ratios or sensational claims.

Common Questions Reveal Patterns of Curiosity

Final Thoughts

Why do people ask this question again and again? Common behavior shows users want clarity, not excitement—only straightforward answers tied to relatable data.

• How do the odds compare across colors?
  Red balls dominate due to higher quantity, increasing likelihood of one appearing—but only a balanced mix yields “all different.”
• *What changes if the box has fewer balls total or different