Consider the Z-channel given in the figure. The input is 0 or 1 with equal probability.

F1 S.B Madhu 27.07.20 D8

If the output is 0, the probability that the input is also 0 equals _________

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GATE EC 2014 Official Paper: Shift 4
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Answer (Detailed Solution Below) 0.79 - 0.81

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Concept:

Bayes’ Theorem is stated as:

p(BA)=p(AB).p(B)p(A) 

Calculation:

P(x=0) = p(x = 1) = 0.5

There are two cases to get output 0:

If the input is zero then output is zero with probability 1, i.e.

p(y=0x=0)=1 

If the input is one the output is zero with probability 0.25, i.e.

p(y=0x=1)=0.25 

Net probability to get output 0 will be:

p(y=0)=p(y=0x=0)×p(x=0)+p(y=0x=1)×p(x=1) 

p(y=0)=1×0.5+0.25×0.5=0.625 

Using Bayes’ Theorem, we can write:

p(x=0y=0)=p(y=0x=0)p(x=0)p(y=0) 

=1×0.50.625=0.8

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