For the COVID example we did in class, say that the entire U…

For the COVID example we did in class, say that the entire US population size is 10000. Let’s change the test accuracy to 96%, i.e. P(T|D)=P(T^c|D^c)=0.96. 1200 people have the disease (12% from our example), 1152 people have the disease and test positive (96% true positive rate). What is P(D|T), that is the probability of having the disease given that you test positive using the Bayes’ rule? Use the tree diagram. 

Let’s say that you saw dark clouds today. There is a 70% cha…

Let’s say that you saw dark clouds today. There is a 70% chance of seeing dark clouds today given that it will rain tomorrow (from historical data). There is a 20% chance of seeing dark clouds today given that it won’t rain tomorrow (again from historical data). There is a 50% chance of seeing dark clouds regardless of whether or not it will rain. The current knowledge without any extra information is that there is a 30% chance of rain tomorrow. What is the updated probability of rain tomorrow given that we saw dark clouds today? Hint: Let A be the event that it will rain tomorrow and B be the event that you observed dark clouds today. You need to find P(A|B). Use the tree diagram and Bayes theorem.