In an army camp, there are 3 soldiers. for each soldier there is a probability of $0.2$ that they have a certain disease (independently from each other). the doctor wants to determine whether any of them have it. since the test is very expensive, he pools together blood samples from all the soldiers. if any of them, i. e., whether any one soldier or any combination of them, have the disease the test is positive with probability $0.9$. if none have it, then the test is positive with probability $0.1$. \begin{enumerate} \item one of the soldiers is called joe. what is the probability that joe has the disease if the test is positive? \item we now change the probabilistic model to take into account that the fridge of the camp also impacts the results of the test. if it is broken, which happens with probability $0.4$, then the test always reads positive. now, what is the probability that that joe has the disease if the test is positive?