To screen passengers for metal, airport security employs a less than perfect metal detector. The good news is that, if someone carries metal, the detector always rings the alarm (with probability 1). However, when a "metal free" person passes through, the detector will annoyingly still sound a false alarm with probability 1/4. Hence, when passengers trigger the alarm they are often asked to go through again. It has been established statistically that 1 out of n passengers carries metal. a) (5) When a passenger arrives and passes through, what is the probability that the detector will sound the alarm? b) (5) If the passenger triggers the alarm, what is the probability that he or she actually carries metal? c) (5) What is the probability that a metal-free passenger clears the detector within at most three attempts? d) (10) How many consecutive times should a passenger trigger the alarm for security agents to conclude that the probability that she or he carries metal is greater than 1/2?