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Consider a Binary Symmetric Channel (BSC) with probability of error being p. To transmit a bit, say 1, we transmit a sequence of three 1s. The receiver will interpret the received sequence to represent 1 if at least two bits are 1. The probability that the transmitted bit will be received in error is

1. p³ + 3p²(1-p)
2. p³
3. (1-p)³
4. p³ + p²(1-p)

Correct answer: p³ + 3p²(1-p)

Solution

The correct option accounts for the scenarios where the transmitted sequence of three 1s is received with at least two bits still being 1. This includes the case where all three bits are received correctly (probability (1-p)³) and the cases where exactly two bits are received correctly and one is flipped (which can occur in three different ways, hence the factor of 3), leading to the total probability of error being p³ + 3p²(1-p).

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