In binary computer transmission, a computer can transmit one of two different si

Question

In binary computer transmission, a computer can transmit one of two different signals (bits), commonly referred to as 0 and 1. The transmission is not error-free and a transmitted 0 can be received as a 1 while a transmitted 1 can be received as a 0. Lets suppose that 63% of all bits transmitted are 0s. 93% of all transmitted 0s are received as 0s while 7% are received as 1s. Similarly, 98% of all transmitted 1s are received as 1s while 2% are received as 0s. Suppose a random bit is transmitted: Let X be the bit received.

a. What is the probability that the transmitted bit is a 1?  
b. What is the probability that X is received as a 1?  
c. Given that a 1 is received, (X=1) what is the probability a 1 was transmitted?  
d. What is the probability that X is received as a 0?  
e. Given that a 0 is received, what is the probability that X was transmitted as a 1?  
f. Suppose 5 bits are independently transmitted. What is the probability they are all zeros?

Explanation / Answer

Let

Z, I = a 0, 1 is transmitted, respectively
z, i = a 0, 1, is received, respectively

a)

P(I) = 1 - P(Z) = 1 - 0.63 = 0.37 [ANSWER]

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b)

By Bayes' Rule,

P(i) = P(Z) P(i|Z) + P(I) P(i|I) = 0.63*(1-0.93) + (1-0.63)*0.98 = 0.4067 [ANSWER]

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c)

Hence,

P(I|i) = P(I) P(i|I)/P(i) = (1-0.63)*0.98/0.4067 = 0.891566265 [ANSWER]

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d)

P(z) = 1 - P(i) = 1 - 0.4067 = 0.5933 [ANSWER]

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e)

P(I|z) = P(I) P(z|I)/P(z) = (1-0.63)*0.02/0.5933 = 0.012472611 [ANSWER]

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f)

As P(Z) = 0.63, then 5 zeroes transmitted is

P = 0.63^5 = 0.099243654 [ANSWER]

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