A scientist observes a rare species of butterfly that appears in 3 distinct colors: red, blue, and green. If she captures 6 butterflies at random, what is the probability that she captures at least one butterfly of each color? Assume each butterfly has an equal chance of being any of the three colors.

Fri, 17 Apr 2026 05:50:13 +0000

A scientist observes a rare species of butterfly that appears in three distinct colors: red, blue, and green. If she captures six butterflies at random, what is the probability she records at least one of each color? This question intrigues nature enthusiasts and data lovers alike—particularly in a digital climate where curiosity about biodiversity and probability puzzles spreads quickly across US mobile users. With growing interest in citizen science and random pattern recognition, understanding how rare color variants emerge in real populations speaks to broader questions in ecology and chance. Rarity and diversity in nature capture public attention, especially when tied to scientific observation. Recent trends in environmental awareness and accessible field research have amplified interest in species exhibiting rare traits—like butterflies displaying unexpected color variations. Social media platforms and educational apps increasingly feature probabilistic puzzles, making this butterfly probability example both relatable and timely. While not a new phenomenon in biology, the blend of ecology with chance calculation offers fresh content potential. To explore the probability, consider a 6-butterfly sample captured at random, each independently having an equal 1/3 chance of being red, blue, or green. The goal is calculating the likelihood of capturing **at least one** butterfly of **each color** in those six trials. Rather than listing every possible combination, mathematicians use complementary counting and combinatorial logic. Begin by computing the total number of equally likely color outcomes: 3⁶, since each butterfly has three color choices. Then, evaluate all combinations that **exclude at least one** color entirely. Subtract these “unfavorable” cases from the total. This process uses inclusion-exclusion to avoid double-counting scenarios where two colors vanish. For instance, eliminate outcomes with only red and blue, only red and green, or only blue and green. By calculating these, scientists estimate the share of samples missing one or more colors.