Paragraph Styles 6. Here is Bayes' Rule: P(Q )= [P(Q) * P(GIO)] / ([P(Q) x P(GIO)] + [P(not Q) x P(Glnet O)D Use Bayes' Rule, to solve the following problem: It is late afternoon in Mayberry. Danny has just robbed the First Bank of Mayberry, the largest bank in town. He runs down Elm Street, into an alley, up a flight of stairs, and through a door. Two policemen are in hot pursuit. When they open the door at the top of the stairs, they find a long hallway. There are five doors on each side of the hallway and at the end is another door. Just as they are surveying the scene, the door at the end of the hallway opens and a woman enters. The policemen run down the hallway and ask her if she has just seen a man matching Danny's description. She says that she has not, and that Danny could not have come through this door since it was locked until just this moment. She holds up the key. They ask her what is behind the ten other doors on the hallway. She explains that this is a hospital, and they are on the Contagious Diseases floor. All the rooms are occupied by patients with one of two diseases, none of the rooms are locked, and all have windows. Four of the rooms hold patients with an extremely contagious disease. The patients in the other six rooms have a disease that is a little bit less contagious, but much more deadly. The policemen consider their next move. Before they can check any of the rooms, one of them gets a call on his radio. Danny has been located and trapped. He is alone in the men's room of a diner on 8 street. This time there is no chance that he can escape, and what's more, he seems ill He can be heard throwing up and flailing around. The doctor explains the situation to the police. Both diseases are top secret. Four of the rooms are occupied by patients with a disease that has the code name LT452. If the proper precautions are not taken, there is a 50 percent chance that someone who is in a room for a minute or less with one of these patients will contract the disease. The other six rooms are occupied by patients with a disease that the doctor refers to as red-6. There is a 20 percent chance that someone who is in a room with one of these patients for a minute or less will contract red-6. "If Danny has LT452, he'll live, at least for now," she says. "If he as red-6, though, he must be given this protein inhibitor in the next five minutes or he won't make it." She pulls a vial from her pocket and holds it up. "Is there any way to tell which disease he has?" asks one of the policemen "No, at this stage, the observable symptoms are the same for both diseases." "Can we ask each of the patients if a man just ran through the room?" "They're all in comas." "We can check which room has an open window." "All of the windows close automatically. These patients are very contagious." "Can we just give Danny the protein inhibitor? If he has red-6, we'll save his life." "No, if he has LT452, the protein inhibitor will kill him. The only thing that we can do is figure out the probability that he was exposed to red-6 given that he's now sick." (1) What is the probability Danny was exposed to red-6 given that he is now sick