Question

Consider a binary symmetric channel (BSC) with a 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 _________.

Answer 1

The probability that the transmitted bit will be received in error in a binary symmetric channel (BSC) can be calculated using the probability of two or more errors occurring in the received sequence.

Step-by-step explanation:

To determine the probability that the transmitted bit will be received in error, we need to calculate the probability of two or more errors occurring in the received sequence. Let's consider the two possible cases:

1. Case 1: Exactly two errors in the received sequence.

2. Case 2: Exactly three errors in the received sequence.

In Case 1, there are three possible positions for the two errors to occur (positions 1 and 2, positions 2 and 3, and positions 1 and 3), and the probability of each position having an error is p. Therefore, the probability of two errors occurring in any of the positions is 3 * p * p * (1 - p). In Case 2, there is only one possibility where all three positions have errors, and the probability of this happening is p * p * p.

Therefore, the probability of the transmitted bit being received in error is:

P(error) = 3 * p * p * (1 - p) + p * p * p.