gen-define-the-binary-symmetric-channel-and-give-one-short-worke-c1e1

Define the binary symmetric channel and give one short worked example.

# Binary Symmetric Channel (BSC) ## Definition A **Binary Symmetric Channel** is a communication channel model where: - **Input and output** are binary: {0, 1} - Each bit is **flipped with probability p** (crossover probability) - Each bit is **transmitted correctly with probability (1-p)** - Errors occur **independently** for each bit The channel can be visualized as: ``` Input Output 0 ---(1-p)--→ 0 \ / \ p / p \ / / X / \ \ / \ \ / (1-p)\ 1 --------→ 1 ``` ## Worked Example **Setup:** BSC with crossover probability p = 0.1 **Input:** 1011 **Question:** What's the probability of receiving exactly 1011? **Solution:** Each bit must be transmitted correctly: - Bit 1 (1→1): probability = 1-p = 0.9 - Bit 2 (0→0): probability = 1-p = 0.9 - Bit 3 (1→1): probability = 1-p = 0.9 - Bit 4 (1→1): probability = 1-p = 0.9 Since errors are independent: P(receive 1011 | sent 1011) = (0.9)⁴ = **0.6561** **Bonus:** P(receive 1001 | sent 1011) = (0.9)³ × (0.1)¹ = 0.0729 (bits 1,2,4 correct, bit 3 flipped)