According to an article in The New Yorker (March 12, 2007), the Department of Homeland Security in the United States is experimenting with installing devices for detecting radiation at bridges, tunnels, roadways, and waterways leading into Manhattan.
The New York Police Department (NYPD) has expressed concerns that the system would generate too many false alarms.
a. Form a contingency table that cross classifies whether a vehicle entering Manhattan contains radioactive material and whether the device detects radiation. Identify the cell that corresponds to the false alarms the NYPD fears.
b. Let A be the event that a vehicle entering Manhattan contains radioactive material. Let B be the event that the device detects radiation. Sketch a Venn diagram for which each event has similar (not the same) probability but the probability of a false alarm equals 0.
c. For the diagram you sketched in part b, explain why P(A | B) = 1, but P(B | A) < 1.