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A drug manufacturer claims that a certain drug cures a blood disease, on the average, \(80 \%\) of the time. To check the claim, government testers used the drug on a sample of 100 individuals and decided to accept the claim if 75 or more were cured. (a) What is the probability that the claim will be rejected when the cure probability is, in fact, \(0.8 ?\) (b) What is the probability that the claim will be accepted by the government when the cure probability is as low as \(0.7 ?\)

A pharmaceutical company knows that approximately \(5 \%\) of its birth-control pills have an ingredient that is below the minimum strength, thus rendering the pill ineffective. What is the probability that fewer than 10 in a sample of 200 pills will be ineffective?

Statistics released by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 out of every 10 drivers on the road is drunk. If 400 drivers are randomly checked next Saturday night, what is the probability that the number of drunk drivers will be (a) less than \(32 ?\) (b) more than \(49 ?\) (c) at least 35 but less than \(47 ?\)

A pair of dice is rolled 180 times. What is the probability that a total of 7 occurs (a) at least 25 times? (b) between 33 and 41 times inclusive? (c) exactly 30 times?

A commonly used practice of airline companies is to sell more tickets than actual seats to a particular flight because customers who buy tickets do not always show up for the flight. Suppose that the percentage of no-shows at flight time is \(2 \%\). For a particular flight with 197 seats, a total of 200 tickets was sold. What is the probability that the airline overbooked this flight?

The serum cholesterol level \(X\) in 14 -year-old boys has approximately a normal distribution with mean 170 and standard deviation 30 . (a) Find the probability that the serum cholesterol level of a randomly chosen 14-year-old boy exceeds 230 (b) In a middle school there are 300 14-year-old boys. Find the probability that at least 8 boys have a serum cholesterol level that exceeds 230 .

A telemarketing company has a special letter opening machine that opens and removes the contents of an envelope. If the envelope is fed improperly into the machine, the contents of the envelope may not be removed or may be damaged. In this case we say that the machine has "failed." (a) If the machine has a probability of failure of 0.01 , what is the probability of more than 1 failure occurring in a batch of 20 envelopes? (b) If the probability of failure of the machine is 0.01 and a batch of 500 envelopes is to be opened, what is the probability that more than 8 failures will occur?

In a certain city, the daily consumption of water (in millions of liters) follows approximately a gamma distribution with \(\mathrm{Q}=2\) and \(\beta=3\). If the daily capacity of that city is 9 million liters of water, what is the probability that on any given day the water supply is inadequate?

Use the gamma function with \(y=\sqrt{2 x}\) to show that \(\Gamma(1 / 2)=\sqrt{\pi}\).

Suppose that the time, in hours, taken to repair a heat pump is a random variable \(X\) having a gamma distribution with parameters \(\alpha=2\) and \(3=1 / 2\). What is the probability that the next service call will require (a) at most 1 hour to repair the heat pump? (b) at least 2 hours to repair the heat pump?