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📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 Number of non-negative integer solutions: $ \binom{3 + 4 - 1}{3} = \binom{6}{3} = 20 $

📌 So there are 20 valid configurations (binary strings) with exactly 3 non-adjacent 1s.

📌 For each such configuration, the probability is:

📌 \left(\frac{1}{3}\right)^3 \left(\frac{2}{3}\right)^5 = \frac{1^3 \cdot 2^5}{3^8} = \frac{32}{6561}

📌 Since each configuration has the same probability and they are mutually exclusive:

📌 Total probability:

📌 \cdot \frac{32}{6561} = \frac{640}{6561}

📌 Thus, the probability is:

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