My question is as follows:
In a nonsymmetric binary channel, $0$ and $1$ are transmitted independently in the proportion of $1:4$. If a $0$ was transmitted, we will receive $0$ with a probability of $0.9$. If a $1$ transmitted, we will receive $1$ with a probability of $0.95$.
a) What is the probability that a received symbol is "$1$"?
b) A "$1$" has been received what is the probability that a "$1$" actually been transmitted?
c) A "$0$" has been received what is the probability that a "$0$" actually been transmitted?
I found below answers but I am not sure:
a)
$P(R_1)=\frac{1}{5}*P(T_0|R_1)+ \frac{4}{5}*P(T_1|R_1)$=$\frac{1}{5}*0.1+ \frac{4}{5}*0.95=0.78$
$(a)$ looks good
Hint for $(b)$
Use Bayes' Theroem:
$$\begin{align*} P(\text{1 transmitted | 1 received}) &=\frac{P(\text{1 transmitted} \cap 1\text{ received})}{P(\text{1 received})}\\\\ \end{align*}$$
where $P(\text{1 received})$ is what you solved correctly in part $(a)$
Hint for $(c)$
This calculation is similar noting that $$P(\text{0 received})=1-P(\text{1 received})$$