Can I apply bayes theory in any way to this problem?

Question

There are two types of reviews: good and bad. A good review has a probability of .3 of being true and a bad review has a probability of .7 of being true. Someone receives 40 good reviews and 10 bad reviews. Given their reviews, what is the probability that this person is of good quality?

Answer 1

Let $P(Q=g)$ be the probability that the quality is good.

$P(Q=g|R=g)$ be the probability that the quality is good, when the review is good.

$P(Q=g|R=b)$ be the probability that the quality is good, when the review is bad.

$P(R=g)$ be the probability that the review is good.

Then, you can write the following

$P(Q=g)=P(R=g) \times P(Q=g|R=g)+ P(R=b) \times P(Q=g|R=b)$

Notice that you can use the information to find $P(R=g)=\frac{40}{50}$ and $P(R=b)=\frac{10}{50}$. Also, $P(Q=g|R=g)=0.3$ and $P(Q=g|R=b)=1-P(Q=b|R=b)=0.3$

Answer 2

I notice that after either review the probability of that person being of good quality appears to be the same: \begin{equation} P(Good|+) = 1-P(Bad|-)=P(Good|-)=0.3 \end{equation} This leads me to believe that after any number of reviews your probability remains unchanged: \begin{eqnarray} P(Good) &=& P(Good|+)P(+)+P(Good|-)P(-)\\&=&P(Good|+)P(+)+P(Good|-)(1-P(+)) \\ &=&P(Good|+)(P(+)+(1-P(+)))\\&=&P(Good|+)\\&=&30\% \end{eqnarray}

You might notice the more general result, which is that if the false negative rate is the same as the true positive rate then the either result gives you no additional information for binary tests on binary systems.