Four large water pumps supply Bellingham, Washington, with water. If water pump i (i 5 1, 2, 3, 4) breaks down, then the time to repair it has an exponential distribution with l 5 1/(2i) hours. Suppose the water pumps operate independently, and when breakdowns occur, the repair times are also independent.
a. Suppose water pump 1 breaks down. What is the probability that it will take more than 30 minutes to repair?
b. Answer part (a) for water pumps 2, 3, and 4.
c. Suppose all four water pumps break simultaneously. What is the probability that at least one of the four water pumps will not be repaired within one hour?
From the instant a fresh-baked chocolate-chip cookie is taken out of the oven, the time (in minutes) the wonderful aroma lasts has an exponential distribution with l = 1/30. Suppose a chocolate-chip cookie is done baking and is taken out of the oven.
a. Find the mean, variance, and standard deviation for the time the aroma lasts .
b. What is the probability that the aroma lasts for at least 40 minutes?
c. What is the probability that the aroma lasts for between 30 and 50 minutes?
d. Find a value t such that the probability that the aroma lasts for at most t minutes is 0.90.
e. Suppose a second batch of cookies is taken out of the oven 10 minutes after the first. What is the probability that there will be no aroma from either batch 35 minutes after the first batch was done?