A communications system consists of two transmitters and receivers. The transmit

Question

A communications system consists of two transmitters and receivers. The transmitters transmit bytes consisting of 8 independent bits, which can each be a 0 or a 1. In Transmitter A, the probability that a bit is a 1 is 0.5. While in Transmitter B, the probability that a bit is a 1 is 0.8. Transmitter A sends three times as many bytes as Transmitter B. The receiver receives bytes from both transmitters.

a. Calculate the probability that a received byte has exactly five 1’s.

b. If a byte has exactly five 1’s, what is the probability that it was sent from Transmitter B?

Explanation / Answer

a) 3T1=T2

the receiver will receive the byte either from transmitter 1 or transmitter 2

if it recive the byte which contains 8 bits from transmitter1 the probility that it will containt ecxactly five 1's = 3(0.5*0.5*0.5*0.5*0.5) = 0.09375

if it recive the byte which contains 8 bits from transmitter2 the probility that it will containt ecxactly five 1's =

0.8*0.8*0.8*0.8*0.8 = 0.32768

the probabity that receiver will receive exactly five 1's will be either from transmitter 1 or transmitter 2 which will be equal to= 0.09375+0.32768 = 0.42413 or 0.42

b) the probabity the the five 1's is from transmitter B= probabilty of transmitter B sending 1/ total probability of receiving five 1's= 0.32768/0.42413=0.851659 or 0.85 or there is 85% probability that it is sent by B

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