A robbery has been committed in an isolated town. Witnesses all agree that the criminal was driving a red pickup truck and had blond hair. Evidence at the scene indicates that the criminal also smoked cigarettes. Police determine that 1/50 of the vehicles in town are red pickup trucks, 30% of the residents smoke, and 20% of the residents have blond hair. The next day they notice a red pickup truck whose driver has blond hair and is smoking. They arrest the driver for the robbery.
a. Is it reasonable to assume that whether someone drives a red pickup truck, smokes cigarettes, and has blond hair are all independent traits? Explain.
b. Assuming that the traits listed are all independent, what proportion of the vehicle owners in town are red pickup truck owners who smoke and have blond hair?
c. Continuing to assume the traits listed are independent, if there are 10,000 vehicle owners in town, how many of them fit the description of the criminal?
d. Assuming that there is no other evidence, what is the probability that the driver arrested by the police is innocent of the robbery?
e. The prosecuting attorney argues that the probability that someone would own a red pickup truck and smoke cigarettes and have blond hair is very small, so the person arrested by the police must be guilty. Do you agree with this reasoning? Explain.