Reference no: EM132854941

The United States Postal Service (USPS) is piloting an unmanned drone delivery system where an operator loads single small package onto a drone, the drone flies to a specified address, drops off the package, and returns to the operator to load another. USPS deems the delivery a "hit" if the drone delivers a package to the correct location and a "miss" if not. After a 1-year trial phase, USPS has determined that if it's raining, the drone hits 80% of the packages it delivers. If it's sunny, the drone never misses. Suppose that they are now running a follow-up phase in an area that rains frequently about 75% of the time, otherwise it is sunny.

(a) What is the probability of a drone hit in this area (accurate delivery of a package)?

(b) If the drone delivers five packages in this area, what is the probability the drone hits exactly four times?

(c) What is the probability the first drone delivery hit occurs on the fourth delivery?

(d) Suppose the number of packages (N) the operator needs to deliver each hour follows a probability mass function pN (n) = 0.4 - 0.1n for n = 0, 1, 2, 3. What is the average and variance of the number of packages that the operator will need to deliver each hour?

(e) Suppose that the delivery times of packages are iid exponential random variables with a mean of 10 minutes (includes loading, delivery, and return time). Also, assume that the operator spends a deterministic amount of time, 40 mins, before starting deliveries (morning team meeting and reviewing protocol). If the operator loads 50 packages for drone deliveries during a shift, what is the probability the shift lasted more than 600 minutes? (Hint: Assume that 50 is "large enough" to make an approximation)