Disease test posterior plus a binomial drug trial

(a) A disease affects $2\%$ of people. A test misses $10\%$ of true cases (false negatives) and flags $1\%$ of healthy people (false positives). Given a positive result, what is the chance the person has the disease?

(b) A lab claims a new drug cures $80\%$ of patients. A regulator treats $20$ patients and counts cures $X$, testing $H_0:p=0.8$ against $H_a:p<0.8$ and rejecting when $X\le12$. What is the significance level?

(c) For the same test, what is the power if the true cure rate is $0.6$?

(a) About $0.648$. (b) About $0.032$. (c) About $0.584$. (a) By Bayes, $\tfrac{0.02\cdot0.9}{0.02\cdot0.9+0.98\cdot0.01}=\tfrac{0.018}{0.0278}$. (b) $\mathbb{P}(X\le12)$ under $\text{Bin}(20,0.8)$. (c) $\mathbb{P}(X\le12)$ under $\text{Bin}(20,0.6)$.