Suppose there is a noisy communication channel in which either $a$ $0 $ or $a $ $1 $ is sent with the following probabilities:
• Probability a $0$ is sent is $0.4.$
• Probability a $1$ is sent is $0.6.$
• Probability that due to noise, a $0$ is changed to a $1$ during transmission is $0.2.$
• Probability that due to noise, a $1$ is changed to a $0$ during transmission is $0.1.$
Suppose that a $1$ is received. What is the probability that a $1$ was sent?
My attempt:
Let $p(b)$ be a $1$ is received and $p(b)$ be a $1$ is sent.
We want to find $p(b \text{ given that } a)$
So we can use bayes rule and we have $$p(a \text{ given that } b) \cdot \frac{p(b)}{p(a)}$$
$P(b)$ is $.6$?
$P (a \text{ given that } b) 1 -.2$?
$P(a) .4 \cdot .2 + .4 \cdot (1 -.2)$ ?
then just fill it in?