(1) A local ice-cream store receives an average of 3 customers per hour. Assume that the arrival events are well described by Poisson distribution. The cashier wants to leave the store to run an errand. The manager computes the expected time T between the arrivals of consecutive customers over the previous week and allows the cashier to leave the store for a duration T. How long does the cashier have to run his errand?

Hint: If the occurrence of events is described by Poisson distribution, then the inter-event
time is described by exponential distribution.

(2) A basketball player can score a basket on fifth attempt, on an average. What is the probability that the player scores the third basket on the tenth attempt?
Hint: Sum of geometric random variables is a negative binomial random variable.

(3) 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 expo- nential 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. Γ(n + 1) = n! if n is a non-negative integer.