A modem transmits over an error-prone channel, so it repeats every "0" or "1" bit transmission five times. We call each such group of five bits a "codeword." The channel changes an input bit to its complement with probability p = 1/10 and it does so independently of its treatment of other input bits. The modem receiver takes a majority vote of the five received bits to estimate the input signal. Find the probability that the receiver makes the wrong decision.

Question

Mathematics

Cesar Mcmillan

12 July, 19:19

A modem transmits over an error-prone channel, so it repeats every "0" or "1" bit transmission five times. We call each such group of five bits a "codeword." The channel changes an input bit to its complement with probability p = 1/10 and it does so independently of its treatment of other input bits. The modem receiver takes a majority vote of the five received bits to estimate the input signal. Find the probability that the receiver makes the wrong decision.

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Celeste Carlson · Answer 1

0.00111

Step-by-step explanation:

Since the modem receiver takes a majority vote of the five

received bits to estimate the input signal, it will only make a wrong decision if 3, 4 or 5 of the 5 bits received are wrong.

Given that the channel changes an input bit to its complement with probability p = 1/10 and it does so independently of its treatment of other input bits, the probability of changing 3 bits out of five is 0.1*0.1*0.1 = 0.001, of changing 4 is 0.0001 and of changing 5 is 0.00001

So, the probability that the modem makes a wrong decision is 0.001+0.0001+0.00001 = 0.00111