Question

Assume that the probability of error-free transmission of a message over a communication channel is p. If a message is not transmitted correctly, a retransmission is initiated. This procedure is repeated until a correct transmission occurs. Such a channel is often called a feedback channel. Assuming that successive transmissions are independent, what is the probability that: a) No retransmission is required. b) Exactly two retransmissions are required

Answer 1

The probability of no retransmission is p, and the probability of exactly two retransmissions can be calculated using the binomial probability formula.

Step-by-step explanation:

To find the probability that no retransmission is required, we need to consider that each transmission has a probability of success, denoted by p. The probability of no retransmission is equal to the probability of the first transmission being successful, which is p. So the probability of no retransmission is p.

To find the probability that exactly two retransmissions are required, we need to consider that each transmission has a probability of failure, denoted by q = 1 - p. The probability of exactly two retransmissions can be calculated using the binomial probability formula. The formula is:

P(X=k) = C(n, k) * p^k * q^(n-k)

Where:

• P(X=k) is the probability of exactly k successes
• C(n, k) is the number of combinations of n items taken k at a time
• p is the probability of success
• q is the probability of failure
• n is the total number of trials

In this case, we want exactly two retransmissions, so k = 2. The total number of trials is not specified, so let's assume it's denoted by n. The probability of exactly two retransmissions is then given by:

P(X=2) = C(n, 2) * p^2 * q^(n-2)