A sump pump in a basement has two battery-operated back-up pumps. In the case of power outage the main sump pump shuts off and the first battery-operated back-up pump kicks in. After the battery powering the first back-up pump runs out of charge, the second battery-operated back-up pump is activated.
Assume that the duration for which a battery can power its pump is described by exponential distribution. The variation in the time the battery lasts is due to the variation in the rate of accumulation of ground water. A battery can power its pump for 20 minutes on an average, before it runs out of charge. A certain power outage lasts an hour. What is the probability that the basement will not get flooded during the power outage?
Hint: You will need to use the Gamma distribution. The integral involved is simple. ?(n + 1) = n! if n is a non-negative integer.