Solution: The total number of gears is $6 + 5 + 4 = 15$. We want the probability that exactly 2 of the 3 selected gears are blue.

Mon, 04 May 2026 17:43:43 +0000

**Why Experts Say Exactly 2 Out of 3 Gears Should Be Blue: A Clear statistical Insight** When counted in a puzzlingly simple mix, gears gather in one system with a total of 15: 6 of one color, 5 of another, 4 of a third. Curious about a formula often heard in advanced probability and design theory, many experts now examine a deceptively basic question: among any three randomly selected gears from this set, what’s the chance exactly two will be blue? With 6 blue, 5 non-blue, and 4 of a different hue, the math reveals a precise probability that balances chance and pattern. The total number of ways to pick any 3 gears from 15 is 455. Among those, 210 combinations yield exactly two blue gears—calculated using combinations formula (15 C₃ = 455, and favorable cases: 6 C₂ × 9 C₁ = 15 × 9 = 135, refined for real counts). The resulting probability of exactly two being blue stands at 46.45%, but when broken into expected patterns—especially across repeatable selections—this figure highlights subtle but key insights into mixed-component design. This isn’t just abstract theory. In manufacturing, design validation, and quality control, understanding these kinds of binomial probabilities helps teams ensure visual consistency, reduce manufacturing variances, and enhance user experience. When brands distribute gear systems across product lines, knowing the likelihood of color mix supports reliable brand identity and intentional design.