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Home » Youll Never Guess Why FM Radio Apps Are Taking Over the World! - Decision Point

Youll Never Guess Why FM Radio Apps Are Taking Over the World! - Decision Point

📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 Question: A robotics engineer is testing a robot that moves on a grid, making exactly 8 steps from the origin $(0,0)$ to some point $(x,y)$, with each step either increasing $x$ by 1 or $y$ by 1. The robot must end at a point where $x + y = 8$, $x \geq 3$, and output $x$ if $x \geq 4$, or $y$ if $x < 4$. What is the expected value of the output?

📌 Solution: The robot takes 8 steps, each independently choosing to go right (increase $x$) or up (increase $y$), uniformly at random (assume fair walk). The endpoint is $(x, 8 - x)$ for $x = 0,1,\dots,8$, so $x$ follows a binomial distribution: $x \sim \text{Binomial}(8, \frac{1}{2})$.

📌 The robot outputs:

📌 $x$, if $x \geq 4$,

📌 $y = 8 - x$, if $x < 4$.

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

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

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

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