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Benton Harbor, Michigan is extraordinary in its high crime and poverty rates. Its police department estimates that, among the male residents age 15+,...
Benton Harbor, Michigan is extraordinary in its high crime and poverty rates. Its police department estimates that, among the male residents age 15+, 1% always carry a loaded gun. The police department has only one public entrance, and it has a metal detector and an armed guard. All persons visiting the police department are asked to remove all the metal objects from their person and step through the metal detector (or inform the guard that they have surgically implanted metal, and submit to a pat-down).
A metal detector makes a loud noise—"goes off"—whenever the machine detects metal, and it's very good at detecting the kinds of alloys used in gun manufacturing, so that in practice the machine goes off whenever a person carrying a gun passes through it. For people who don't carry a gun, and who meticulously remove other metals from their person before passing through the metal detector, and who don't have any metal in their bodies due to surgery, the machine doesn't go off at all. However, in practice, some people tend to forget to remove all metals from their person before stepping through the metal detector, or to inform the guard of surgically implanted metal. So in practice, for people who don't carry a concealed weapon, about one time in 100 the machine will still go off.
a) From the perspective of the armed guard: an adult man walks in, places some car keys and change in the tray provided for metal objects, and steps through the metal detector. The metal detector goes off. Which is more likely: that the person is carrying a gun, or that the person is not carrying a gun? Present your answer, and, as always, present evidence to back up your claim.
b) State the key assumptions not explicitly stated in the description of the problem that you relied on to get your result in part (a), and briefly explain to what extent each assumption is reasonable or realistic.