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📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 An ichthyologist studying fish populations categorizes each fish by size (small, medium, large) and species (trout, salmon, bass). If she observes 4 fish, each independently selected with equal likelihood across all categories, what is the probability that she observes at least one fish of each species?

📌 There are \(3\) species, so the total number of possible species assignments for 4 fish is:

📌 We want the number of outcomes where all three species appear. Use complementary counting: subtract cases missing at least one species.

📌 Let \( A \), \( B \), \( C \) be the events that trout, salmon, and bass are missing, respectively.

📌 \( |A| = |B| = |C| = 2^4 = 16 \) (only two species used)

📌 \( |A \cap B| = |A \cap C| = |B \cap C| = 1^4 = 1 \) (only one species used)

📌 \( |A \cap B \cap C| = 0 \) (impossible)

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