Binary Symmetric Channel (Independent Bit Errors): For a 3-bit block with independent bit error probability p, find the probability of at most one bit error.

Introduction / Context:
In digital communications, bit errors on a memoryless channel are often modeled as independent Bernoulli events with error probability p. For small frames or blocks, exact probabilities for events like “at most one error” are calculated with the binomial distribution, guiding the design of coding and error detection.

Given Data / Assumptions:

• Block length n = 3 bits.
• Each bit error is independent with probability p (0 ≤ p ≤ 1).
• We seek P(X ≤ 1), where X = number of bit errors in the block.

Concept / Approach:

Use the binomial distribution: P(X = k) = C(n, k) p^k (1 − p)^(n − k). Then sum for k = 0 and k = 1 to get “at most one error”.

Step-by-Step Solution:

Verification / Alternative check:

Complement method: P(X ≥ 2) = P(2) + P(3) = 3 p^2 (1 − p) + p^3. Then 1 − P(X ≥ 2) = 1 − [3 p^2 (1 − p) + p^3] simplifies to (1 − p)^3 + 3 p (1 − p)^2, confirming the result.

Why Other Options Are Wrong:

• 3 p^2 (1 − p) + p^3: This is P(X ≥ 2), the complement of the desired probability.
• 1 − [(1 − p)^3 + 3 p (1 − p)^2]: This equals P(X ≥ 2), not “at most one”.
• (1 − p)^3: Only the zero-error case; missing single error events.
• 1 − p^3: Not a binomial sum for n = 3; incorrect.

Common Pitfalls:

• Forgetting to include exactly one error in the “at most one error” event.
• Mixing complement incorrectly (using 1 − P(X ≤ 1) rather than 1 − P(X ≥ 2)).

Final Answer:

(1 - p)^3 + 3 p (1 - p)^2