A digital communication channel transmits bits of information, usually designated as 0 and 1. If a sender transmits a 0, he/she hopes the recipient receives a 0. If a sender transmits a 1, he/she hopes that the recipient receives a 1. Unfortunately, this is not always the case. Suppose on a certain transmission line, a transmitted 0 is received correctly 90% of the time (10% of the time a 1 is received), and a transmitted 1 is received correctly 74% of the time. (26% of the time a 0 is received.) It is known that on this transmission line, 80% of all bits transmitted are 0 bits. A randomly selected transmitted bit is examined. Say its value is X. The bit received is Y. a.What is the probability that X = 0?  b.What is the probability that X = 0 and Y = 0?  c. What is the probability that X = 1 and Y = 0?  d. What is the probability that Y = 0?  e. What is the probability that X = 0 given that Y = 0?

A First Course in Probability (10th Edition)
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Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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A digital communication channel transmits bits of information, usually designated as 0 and 1. If a sender transmits a 0, he/she hopes the recipient receives a 0. If a sender transmits a 1, he/she hopes that the recipient receives a 1. Unfortunately, this is not always the case. Suppose on a certain transmission line, a transmitted 0 is received correctly 90% of the time (10% of the time a 1 is received), and a transmitted 1 is received correctly 74% of the time. (26% of the time a 0 is received.) It is known that on this transmission line, 80% of all bits transmitted are 0 bits.
A randomly selected transmitted bit is examined. Say its value is X. The bit received is Y.
a.What is the probability that X = 0? 
b.What is the probability that X = 0 and Y = 0? 
c. What is the probability that X = 1 and Y = 0? 
d. What is the probability that Y = 0? 
e. What is the probability that X = 0 given that Y = 0? 
f. If 1000 random transmitted bits are examined, what is the expected number of 0 bits in this group. (The transmitted bits) 
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