Assume that the probability of error-free transmission of a message over a communication channel is . if a message is not transmitted correc

Question

Assume that the probability of error-free transmission of a message over a communication channel is . 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. assume that successive transmissions are independent.
a. what is the probability that no retransmissions are required? (note that the number of retransmissions is the number of transmissions minus one.)

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    2021-09-14T22:08:27+00:00

    The geometric distribution applies to experiments employing Bernoulli trials (success or failure) to determine the number of trials to obtain the first success.

    This distribution may be applied to the given situation, where the probability of success is p.

    Let k=number of retransmissions required.

    (a) no retransmissions, k=0
    If no retransmissions are required, it means that the transmission is sucessful on first try, with probability
    P(k)=P(0)=p.

    Note: for k retransmissions (failures), the probability is
    P(k)=(1-p)^k*p

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    2021-09-14T22:08:39+00:00

    Answer:

    Let the probability of error-free transmission of a message over a communication channel is P.

    Then the Probability of Retransmission  is =1-P

    Now the question is Probability when no retransmission is required

                                = Probability of error free transmission in first attempt

                                   =P

    Now , suppose after first failure error free transmission takes place.

    Since events are independent.

    ∵P[error free transmission after a failure ]= P[1-P]

    If after two attempts error free transmission takes place

    then P[ error free transmission after two failures]= P [1-P]²

    So, For after n failures

    P[error free transmission after n failure]=P[1-p]^n

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