20. A certain port, cargo on large ships is transferred to a pier using a smaller craft, which cycles back and forth between the ship and the store. The time between arrivals of the smaller craft at the pier has an:
Mean: 25 mins.
Suppose there is only one unloading zone at the pier available for the small craft to use. If a craft docks at 9 a.m. and doesn't finish unloading until 9:20 a.m., what is the probability that a second craft will arrive during this time at the unloading zone and have to wait before docking?
Solution: Letbe the time the second craft takes to arrive. We know thathas an exponential distribution with parameter. Therefore, the probability that a second craft will have to wait before docking is
21. The amount of soda a dispensing machine pours into a 12 oz. can of soda:
Standard deviation: 0.02
Every can that has more than 12.05 ozs. of soda poured into it, causes a spill and the can needs to go through a special cleaning process before it can be sold. What is the mean amount of soda the machine should dispense if the company wants to limit the percentage that need to be cleaned because of spillage to 3%?
Solution: Letbe the amount of soda dispensed by the machine. We need
Using the inverse cumulative standard normal distribution we get
and solving forwe find that