# Thread: Digital Communication Question - Conditional Probability

1. ## Digital Communication Question - Conditional Probability

A digital communications system consists of a transmitter and a reciever. During each short transmission interval the transmitter sends a signal which is to be interpreted by a zero, or it sends a different signal which is to be interpreted as a one. At the end of each interval the reciver makes the best guess at that was transmitted. Consider the events:

T0 = {Transmitter sends 0}
T1 = {Transmitter sends 1}
R0 = {Receiver concludes that a 0 was sent}
R1 = {Reciever concludes that a 1 was sent}

Assume that P(R0|T0) = 0.99, P(R1|T1) = 0.98 and P(T1) = 0.5

Find
a) the probability of a transmission error given R1?
b) the overall probability of a transmission error
c) repeat a) and b) assuming P(T1) = 0.8 instead of 0.5
What I really need to know is how to do a)

I started off by calculating P(T0) = 1 - P(T1)
then I wanted to use the formula

P(T0|R1) = P(T0*R1)/P(R1) [* = intersect]

I calculated the numerator as P(T0*R1) = P(T0)P(R1|T0)

and P(R1|T0) = 1 - P(R0|T0)

but how do I calculate P(R1)

2. $\displaystyle P(R_1)=P(R_1\cap T_0)+P(R_1\cap T_1)=P(R_1|T_0)P(T_0)+P(R_1|T_1)P(T_1)$

3. $\displaystyle P(R_1\cap T_0)+P(R_1\cap T_1)=P(R_1|T_0)P(T_0)+P(R_1|T_0)P(T_0)$

I understand the first part of your equivalence but is this part a typo?

4. Originally Posted by Hasan1
$\displaystyle P(R_1\cap T_0)+P(R_1\cap T_1)=P(R_1|T_0)P(T_0)+P(R_1|T_1)P(T_1)$

I understand the first part of your equivalence but is this part a typo?
Indeed there was. It is correct now.