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 $P\left(T_1\right)$ = $0.4$
$\textbf{hint:}$ $$ P\left(T_1|R_1\right) = 1 $$ 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.