Question: A sump pump in a basement has two battery-operated back-up pumps. In case of power outage the main sump pump shuts off and the first battery-operated pump gets activated. After the first battery runs out of charge the second back-up pump gets activated.

The duration for which a battery’s charge lasts depends on the rate of accumulation of ground water. Assume that a battery can power its pump for 20 minutes, on an average, and that the duration for which a battery’s charge lasts is described by an exponential random variable. If a power outage lasts an hour, what is the probability that the basement will not get flooded during the outage? (Hint: the sum of exponential random variables is a Gamma random variable.)

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