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Hospital administrators in large cities anguish about problems with traffic in emergency rooms in hospitals. For a particular hospital in a large city, the staff on hand cannot accommodate the patient traffic if there are more than 10 emergency cases in a given hour. It is assumed that patient arrival follows a Poisson process and historical data suggest that, on the average, 5 emergencies arrive per hour. (a) What is the probability that in a given hour the staff can no longer accommodate the traffic? (b) What is the probability that more than 20 emergencies arrive during a 3 -hour shift of personnel?

In airport luggage screening it is known that \(3 \%\) of people screened have questionable objects in their luggage. What is the probability that a string of 15 people pass through successfully before an individual is caught with a questionable object? What is the expected number in a row that pass through before an individual is stopped?

Refusal rate in telephone polls is known to be approximately \(20 \%\). A newspaper report indicates that 50 people were interviewed before the first refusal. (a) Comment on the validity of the report. Use a probability in your argument. (b) What is the expected number of people interviewed before a refusal?

During a manufacturing process 15 units are randomly selected each day from the production line to check the percent defective. From historical information it is known that the probability of a defective unit is \(0.05 .\) Any time that two or more defectives are found in the sample of \(15,\) the process is stopped. This procedure is used to provide a signal in case the probability of a defective has increased. (a) What is the probability that on any given day the production process will be stopped? (Assume \(5 \%\) defective.) (b) Suppose that the probability of a defective has increased to \(0.07 .\) What is the probability that on any given day the production process will not be stopped?

A car rental agency at a local airport has available 5 Fords, 7 Chevrolets, 4 Dodges, 3 Hondas, and 4 Toyotas. If the agency randomly selects 9 of these cars to chanffeur delegates from the airport to the downtown convention center, find the probability that 2 Fords. 3 Chevrolets, 1 Dodge, 1 Honda, and 2 Toyotas are used.

Service calls come to a maintenance center according to a Poisson process and, on the average, 2.7 calls come per minute. Find the probability that (a) no more than 4 calls come in any minute; (b) fewer than 2 calls come in any minute; (c) more than 10 calls come in a 5 -minute period.

An electronics firm claims that the proportion of defective units of a certain process is \(5 \%\). A buyer has a standard procedure of inspecting 15 units selected randomly from a large lot. On a particular occasion, the buyer found 5 items defective. (a) What is the probability of this occurrence, given that the claim of \(5 \%\) defective is correct? (b) What would be your reaction if you were the buyer?

An electronic switching device occasionally malfunctions and may need to be replaced. It is known that the device is satisfactory if it makes, on the average, no more than 0.20 error per hour. A particular 5-hour period is chosen as a "test" on the device. If no more than 1 error occurs, the device is considered satisfactory. (a) What is the probability that a satisfactory device will be considered unsatisfactory on the basis of the rest? Assume that a Poisson process exists. (b) What is the probability that a device will be accepted as satisfactory when, in fact, the mean number of errors is \(0.25 ?\) Again, assume that a Poisson process exists.

(a) Suppose that you throw 4 dice. Find the probability that you get at least one 1 . (b) Suppose that you throw 2 dice 24 times. Find the probability that you get at least one \((1,1),\) that is, you roll "snake;-eyes." [Note: The probability of part (a) is greater than the probability of part (b).]

The potential buyer of a particular engine requires (among other things) that the engine start successfully 10 consecutive times. Suppose the probability of a successful start is 0.990 . Let us assume that the outcomes of attempted starts are independent. (a) What is the probability that the engine is accepted after only 10 starts? (b) What is the probability that 12 attempted starts are made during the acceptance process?