Question

16. In a noisy communication system transmitting binary digits (zeros and ones) the sender may send a digit (for example, 1) and the receiver may receive the other digit (for example, 0). Let A be the event "a 1 is sent" and let B be the event "a 1 is received." Hence T is the event "a O is sent" and B is the event "a 0 is received." Assume that Suppose further that PrA-5. Give a physical interpretation of the forego- ing conditional probabilities. Determine the probabilities Pr(AB), Pr(ג PAlB], and PB These conditional probabilities are the ones that the receiver might want to know to help in interpreting a message

Answer 1

here P(B)=P(A)*P(B|A)+P(Ac)*P(B|Ac)=0.5*(1-0.01)+0.5*(0.01)=0.5

hence P(Bc)=1-P(B)=1-0.5=0.5

hence

P(A|B)=P(A)*P(B|A)/P(B)=0.5*(1-0.01)/0.5 =0.99

P(Ac|B)=P(Ac)*P(B|Ac)/P(B)=0.5*(0.01)/0.5=0.01

P(A|Bc)=P(A)*P(Bc|A)/P(Bc)=0.5*0.01/0.5=0.01

P(Ac|Bc)=P(Ac)*P(Bc|Ac)/P(Bc)=0.5*0.99/0.5=0.99