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Home » Check using the Pythagorean theorem: \( 7^2 + 24^2 = 49 + 576 = 625 \). - High Altitude Science

Check using the Pythagorean theorem: \( 7^2 + 24^2 = 49 + 576 = 625 \). - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 We want $ \mathbb{E}[\text{output}] $.

📌 Since $x$ is integer from 0 to 8, define indicator for each possible $x$:

📌 Let $E = \sum_{k=0}^{8} \left( \text{output}(k) \cdot \mathbb{P}(x = k) \right)$

📌 $\mathbb{P}(x = k) = \binom{8}{k} \left(\frac{1}{2}\right)^8 = \frac{\binom{8}{k}}{256}$

📌 For $k = 0,1,2,3$: output = $y = 8 - k$

📌 For $k = 4,5,6,7,8$: output = $k$

📌 \mathbb{E}[\text{output}] = \sum_{k=0}^{3} (8 - k) \cdot \frac{\binom{8}{k}}{256} + \sum_{k=4}^{8} k \cdot \frac{\binom{8}{k}}{256}

📌 Compute each sum.

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