Question

For a certain binary communication channel the probability that a transmitted 0 is received as a 0 is 0.95 and the probability that a transmitted 1 is received as 1 is 0.9. If the probability that a 0 is transmitted is 0.4. Find the probability that i. A 0 is received ii. A 0 was transmitted given that a 0 was received.

Answer 1

P( 0T )= 0.4 i. e probability of 0 transmitted.  

P( 1T ) = 0.6  

P( 1R ) = probability of receiving 1.  

P(1R) = 0.4 x ( 1 - .95 ) + 0.6 x 0.90 = 0.56  

P ( 1T ⋂ 1R ) = 0.6 x 0.9 = 0.54  

P (1 T/ 1R ) = 0.54 / 0.56 = 0.96

Answer 2

is event of transmitting ‘1’, ̅is event of transmitting ‘0’, is event of receiving ‘1’, ̅ is event of receiving‘0’.

Case (i)  : probability.(|) =()(|)/()=0.6 ∙ 0.9.0.56 =27/28.

Case (ii)  : probability.() = ()(|) + (̅)(|̅) = (1 − 0.4)0.9 + 0.4(1 − 0.95) = 0.6 ∙ 0.9 + 0.4 ∙ 0.05= 0.56.  Hence the Answer