# Finding the probability of $R_1$

Consider a binary communication system that consists of a transmitter, a receiver and a Chanel that transfers bits from the transmitter to the receiver. The nature of the channel is such that it occasionally drops bits, so that when a zero or a one is transmitted, it is possible that nothing is received. to simplify the formulation of the model we denote the event of not receiving as $R_e$. We define the following events:

$R_e$ = error in transmission

$T_o$ = transmission of a zero

$T_1$ = transmission of a one $$\begin{eqnarray} P\left(R_o|T_o\right) &=& 0.9,\\ P\left(R_e|T_o\right) &=& 0.1,\\ P\left(R_1|T_1\right) &=& 0.8,\\ P\left(R_e|T_1\right) &=& 0.2. \end{eqnarray}$$ Question: Calculate $P\left(R_1\right)$

My attempt: $$P\left(R_1|T_1\right) =\frac{P\left(R_1\cap T_1\right)}{P\left(T_1\right)}$$

therefore $$P\left(R_1\cap T_1\right) = P\left(T_1\right)P\left(R_1|T_1\right)$$

and that's as far as i can go, is there a way i can separate $P\left(R_1\cap T_1\right)$ to get just $P\left(R_1\right)$

-Thanks!

edit:

The following is also given:

For simpilicity we assume that the transmitterd signal satisfies

$P\left(T_o\right)$=$0.6$, and $P\left(T_1\right)$ = $0.4$

• Hi there, and welcome! I have edited your post this time. But please take a look at this mathjax tutorial. Aug 2, 2014 at 21:57
• As for your question can you fill out the details of the probabilities and indeed the original question? i.e. what is $R_1$? As it stands (my knowledge of Bayes is not complete by any means) but I think some further information is required. Aug 2, 2014 at 21:59
• Thanks for cleaning up my question :). i have edited the initial question and copied it out as it is in my text book. Aug 2, 2014 at 22:07

$\textbf{hint:}$ $$P\left(T_1|R_1\right) = 1$$ I am assuming (due to the level of the problem) that the only events that can occur from a given transmission is either the correct bit or no bit i.e. the $R_e$ event. therefore the probability that the transmitting 1 given that the system received a one is unity.
• $$P\left(R_1|T_1\right) =\frac{P\left(T_1|R1\right)P\left(R_1\right)}{P\left(T_1\right)}$$ is this what you have in mind?.. $$P\left(R_1\right) =\frac{P\left(R_1|T_1\right)P\left(T_1\right)}{P\left(T_1|R_1\right)}$$ Aug 3, 2014 at 7:36