We just had a massive trial with two witnesses: A and B. They both have a long history of not being fully accurate. A has a probability of $\frac{13}{14}$ of telling the truth and B has a probability of $\frac{14}{15}$ of telling the truth. Both A and B stated that convict C was guilty! We know that statistically, C is guilty with probability $\frac{1}{300}$. What is the probability that C is actually guilty? (We can assume that A and B telling the truth are independent events.)
I know that this is an example of Bayes' Theorem. But when I apply the theorem don't the numerator and denominator cancel out leaving $\frac{1}{300}$ as the answer. It seems that I am a bit confused with what exactly we need to calculate. Could someone please clarify mathematically what is asked in the question in reference to the theorem.
Thanks
Edit: My working
Given $P$(guilty)$= \frac{1}{300}$
We need to find $P$(guilty| both stated guilty). This is given by
$\frac{P(both stated guilty| guilty)P(guilty)}{P(both stated guilty)}$ from Bayes' Theorem. At this point, I think the conditional term in the numerator and the denominator are the same and all that remains is $P$(guilty)$= \frac{1}{300}$. I am obviously missing something but I can't figure out what.