USE BAYES LAW to explain reasoning.

1. In a certain town, there are two taxi companies: Green Cabs Ltd. And Blue Taxi Inc. The two companies fleets of cars are all painted green and blue, respectively. Green Cabs dominates the market: 80% of the taxis on the road are green. On a misty winter night, a cab sideswipes a pickup (which crashes) and drives off. Theres a witness who says the sideswiper was blue. On the basis of this testimony, the driver of the pickup files a lawsuit against Blue Taxi Inc. In preparing her defense, Blue Taxis attorney tests the witnesss ability to identify cabs under similar conditions to the night of the crash. They run a number of trials, and he correctly identifies cab color 80% of the time. On this basis, she succeeds in vindicating her clients using Bayes Rule. Whats the probability that the sideswiper is in fact blue, as the witness claimed?

2. A young man goes to the doctor because hes been vomiting. The doctor figures there are three possible explanations: food poisoning, appendicitis, and its just one of those thingsi.e., the patient is healthy but just happened to throw up. Appendicitis is life-threatening so we want to know the probability that the patient has that. The doctor knows the following facts: the rates at which people at this persons age get appendicitis (6 in 100,000) and food poisoning (9 in 100,000), and the frequency with which vomiting occurs when one has appendicitis (80% of the time), when one has food poisoning (90%), and when one is healthy (very rarely: 1 in 100,000).

(a) Whats the probability the patient has appendicitis?

(b) The patient comes back the next day complaining of sharp abdominal pain. The doctor knows that 60% of appendicitis patients show this symptom, but thats relatively rare in cases of food poisoning (only 2%)and almost unheard of for healthy people (1 in 1,000,000). Now whats the probability the patient has appendicitis?