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1. A quality control engineer is in charge of testing whether or not 90% of the DVD players produced by his company conform to specifications. To do this, the engineer randomly selects a batch of 10 DVD players from each day's production. The day's production is acceptable provided no more than 1 DVD player fails to meet specifications. Otherwise, the entire day's production has to be tested.

(i) What is the probability that the engineer incorrectly passes a day's production as acceptable if only 70% of the day's DVD players actually conform to specification?

(ii) What is the probability that the engineer unnecessarily requires the entire day's production to be tested if in fact 95% of the DVD players conform to specifications?

2. Bits are sent over a communications channel in packets of 12. If the probability of a bit being corrupted over this channel is 0.15 and such errors are independent, what is the probability that no more than 3 bits in a packet are corrupted? If 6 packets are sent over the channel, what is the probability that at least one packet will contain 4 or more

corrupted bits?

3. A rat has to choose between 6 doors, one of which contains chocolate. If the rat chooses the wrong door, it is returned to the starting point and chooses again, and continues until it gets the chocolate. Let X be the serial number of the trial on which the chocolate is found.

(a) Find the probability function of X.

(b) What is the expectation of X?

4

The number of visitors to a web server per minute follows a Poisson distribution. If the average number of visitors per minute is 5, what is the probability that

(i) There are two or fewer visitors in one minute?

(ii) There are exactly two visitors in 40 seconds?.

5 The mileage (in thousands of miles) that car owners get with a certain kind of radial tire is a random variable having an exponential distribution with 0 = 40. Find the probabilities that one of these tires will last (a) at least 20,000 miles; (b) at most 30,000 miles. 6 The amount of time that a watch will run without having to be reset is a random variable having an exponential distribution with e = 120 days. Find the probabilities that such a watch will (a) have to be reset in less than 24 days; (b) not have to be reset in at least 180 days