A nonsymmetric binary communications channel sends bit $0$ with probability $p$ and bit $1$ with probability $1-p$
Let $A_i$ be the event that bit $i$ is sent through the channel, $i=\{0,1\}$,
$B_i$ be the event that bit $i$ is received by the channel, $i=\{0,1\}$
a) Find the probability that the output is $0$
b) Find the probability that the input was $0$ given that the output is 1
Given:
$$P[B_0|A_0]=1-\epsilon_1$$
$$P[B_0|A_1]=\epsilon_2$$
$$P[B_1|A_0]=\epsilon_1$$
$$P[B_1|A_1]=1-\epsilon_2$$
a) Is it correct that $P[B_0] = P[B_0|A_0]P[A_0] + P[B_0|A_1]P[A_1] = (1-\epsilon_1)p + \epsilon_2(1-p) = p(1-\epsilon_1-\epsilon_2)+\epsilon_2$
b) My question here: We know that the probability of the output being $1$ given the input was $0$ is $\epsilon_1$. Why isn't this the same as the probability of the input being $1$ given the output is $0$ ?
Best Answer
The two probabilities are different.
Given $P(B_1|A_0)=\epsilon_1$ and $P(A_0)=p$, you need to use the Bayes' theorem to calculate $P(A_0|B_1)$ as follows
$$P(A_0|B_1)=\frac{P(B_1|A_0)P(A_0)}{P(B_1)}$$ and $P(B_1)$ can be calculated similar to $P(B_0)$.