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

Question

6. 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 = transmitted 0
z = received 0
I = transmitted 1
i = received 1

a)

As P(Z) = 0.63

Then

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

**********************

B)

As

P(i) = P(Z) P(i|Z) + P(I) P(i|I) = 0.63*0.07 + (1-0.63)*0.98 = 0.4067 [ANSWER]

*******************

c)

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

*****************

d)

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

*******************

e)

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

*******************************************
F)

Note that P(Z) = 0.63

Hence, for 5 0's,

P(0,0,0,0,0) = 0.63^5 = 0.099243654 [ANSWER]

Related Questions

Navigate

• Browse (All)
• Browse 6
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.