T&T Problem 2.39 In the serial transmission of a byte of information (e.g. over

Question

T&T Problem 2.39 In the serial transmission of a byte of information (e.g. over a network) errors in each bit occur independently with probability q . For every 8-bit byte sent, a parity check bit is appended so that each byte transmitted actually consists of 9 bits. The parity bit is chosen so the group of 9 bits has “even parity”, i.e., the number of 1’s contained in the 9 bit group is even. Errors can be detected by checking the parity of the received 9 bit sequence.

a) What is the probability that a single error occurs in the sequence?

b) What is the probability that the errors occur but are not detected? Write your answer as an expression involving the error probability, q.

Explanation / Answer

2.33 If P(Single bit error) = p

Then

P(Consecutive errors) = p*2p = 2p^2

Given,

2p^2 = 2 x 10^(-4)

p^2 = 10^(-4)

p = 10^(-2)

p = 0.01

2.39 Since error in any bit occur independently. We need to find the probability of single error in a 9 bit sequence. So,

P(Single error) = 9C1 * q^1 * (1 - q)^8 = 9q*(1 - q)^8

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