$A$ speaks the truth with probability $p$. An event $E$ occurs with probability $\alpha$. I have to find the probability of occurrence of $E$ given that $A$ said $E$ occured.
My approach:
$P[E|A_E] = \frac{P[A_E|E]P[E]}{P[A_E|E]P[E] + P[A_E|E^c]P[E^c]}$
Now I have to find $P[A_E|E]$ and $P[A_E|E^c]$ from the following information
$P[A_E|E]P[E] + P[{A_E}^c|E^c]P[E^c] = p$,
$P[A_E|E^c] + P[{A_E}^c|E^c] = 1$,
$P[A_E|E] + P[{A_E}^c|E] = 1$
But I could not find any way. Please help.
If A speaks truth with probability p, if he says E occurred it did occur with probability $\alpha$. If A does not speak the truth with probability (1-p), if he says E occurred then it did not occur with probability $1-\alpha$.
Now $$P(E/\text{A says E occurred}) = \frac{\alpha p }{\alpha p + (1-\alpha)(1-p)}$$