In a printing shop, print requests arrive randomly and independently at an average rate of 20 per hour and are placed in a queue according to arrival times. Suppose that the time it takes to process each request is exponentially distributed and that the print times for different jobs are independent. (a) Calculate the probability that this printer will be idle for the next 5 minutes if the queue is currently empty. (b) A particular printer in this shop is capable of printing five pages per minute. The average request results in ten printed pages of poster. (1) Calculate the probability that the next request will take more than 2 minutes to process. (ii) There is currently one job in the queue and it has been active for the last 5 minutes. Calculate the probability that this job will still be active 1 minute from now. (iii) State the property of the distribution that you can see from the result in (ii).

The life of a certain type of device has an exponentially distributed failure with rate .008 per day. a) What is the mean time to failure? b) What is the probability that 200 days will pass before a failure is observed? c) What is the probability that 4 of the next 6 failures observed will occur after 200 days have passed? d) What is the probability that at most 3 failures will occur in 200 days?

The unemployment rate in a city is 15%. If 6 people from the city are sampled at random, find the probability that fewer than 2 of them are unemployed. Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places. (If necessary, consult a list of formulas.)

A life insurance salesman operates on the premise that the probability that a man reaching his sixtieth birthday will not live to his sixty-first birthday is 0.05. On visiting a holiday resort for seniors, he sells 8 policies to men approaching their sixtieth birthdays. Each policy comes into effect on the birthday of the insured and pays a fixed sum on death. All 8 policies can be assumed to be mutually independent. Provide answers to the following to 3 decimal places. Part a) What is the expected number of policies that will pay out before the insured parties have reached age 61? Part b) What is the variance of the number of policies that will pay out before the insured parties have reached age 61? Part c) What is the probability that at least two policies will pay out before the insured parties have reached age 61?

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Suppose that the login times to a computer network follows a Poisson process with an average of 4 logins per minute. Find the probability of exactly 6 logins in 2 minutes.

In any given year a factory has a 10% probability of having a serious accident about every how Nagy years might the factory expect to have an accident?

A login password consists of 4 letters followed by 2 numbers. Assume that the password is Case-sensitive. What is the probability that randomly chosen password begins with the letters AB?

A sample of 500 students during the fall semester shows that 300 took ECON 201, 220 took MATH 126, and 80 students took both at the same time. What is the probability that a student took ECON 201 or MATH 126.

3)  A light fixture contains five lightbulbs. The lifetime of each bulb is exponentially distributed with mean 200 hours. Whenever a bulb burns out, it is replaced. Let T be the time of the first bulb replacement. Let Xi, i = 1, . . . , 5, be the lifetimes of the five bulbs. Assume the lifetimes of the bulbs are independent.