Page 126 - Applied statistics and probability for engineers

104 Chapter 3/Discrete Random Variables and Probability Distributions

(b) What is the probability that at least one packet must be inspection of purity. If two of the totes do not conform to purity
re-sent? requirements, what is the probability that at least one of the
(c) What is the probability that two or more packets must be nonconforming totes is selected in the sample?
re-sent? 3-187. The probability that your call to a service line is
(d) What are the mean and standard deviation of the number of answered in less than 30 seconds is 0.75. Assume that your
packets that must be re-sent? calls are independent.
(e) If there are 10 messages and each contains 100 packets, (a) If you call 10 times, what is the probability that exactly
what is the probability that at least one message requires nine of your calls are answered within 30 seconds?
that two or more packets be re-sent? (b) If you call 20 times, what is the probability that at least 16
3-181. A particularly long trafic light on your morning calls are answered in less than 30 seconds?
commute is green on 20% of the mornings. Assume that each (c) If you call 20 times, what is the mean number of calls that
morning represents an independent trial. are answered in less than 30 seconds?
(a) What is the probability that the irst morning that the light 3-188. The probability that your call to a service line is
is green is the fourth morning? answered in less than 30 seconds is 0.75. Assume that your
(b) What is the probability that the light is not green for 10 calls are independent.
consecutive mornings? (a) What is the probability that you must call four times to
obtain the irst answer in less than 30 seconds?
3-182. The probability is 0.6 that a calibration of a transducer
in an electronic instrument conforms to speciications for the (b) What is the mean number of calls until you are answered in
measurement system. Assume that the calibration attempts are less than 30 seconds?
independent. What is the probability that at most three calibration 3-189. The probability that your call to a service line is
attempts are required to meet the speciications for the measure- answered in less than 30 seconds is 0.75. Assume that your
ment system? calls are independent.
(a) What is the probability that you must call six times in
3-183. An electronic scale in an automated illing opera- order for two of your calls to be answered in less than 30
tion stops the manufacturing line after three underweight seconds?
packages are detected. Suppose that the probability of an (b) What is the mean number of calls to obtain two answers in
underweight package is 0.001 and each ill is independent. less than 30 seconds?
(a) What is the mean number of ills before the line is stopped?
(b) What is the standard deviation of the number of ills before 3-190. The number of messages that arrive at a Web site is a
the line is stopped? Poisson random variable with a mean of ive messages per hour.
(a) What is the probability that ive messages are received in
3-184. The probability that an eagle kills a rabbit in a day
of hunting is 10%. Assume that results are independent for 1.0 hour?
each day. (b) What is the probability that 10 messages are received in 1.5
(a) What is the distribution of the number of days until a suc- hours?
cessful hunt? (c) What is the probability that fewer than two messages are
(b) What is the probability that the irst successful hunt occurs received in 0.5 hour?
on day ive? 3-191. Four identical computer servers operate a Web site.
(c) What is the expected number of days until a successful hunt? Only one is used to operate the site; the others are spares that can
(d) If the eagle can survive up to 10 days without food (it requires be activated in case the active server fails. The probability that a
a successful hunt on the 10th day), what is the probability that request to the Web site generates a failure in the active server is
the eagle is still alive 10 days from now? 0.0001. Assume that each request is an independent trial. What
is the mean time until all four computers fail?
3-185. Trafic low is traditionally modeled as a Poisson dis-
tribution. A trafic engineer monitors the trafic lowing through 3-192. The number of errors in a textbook follows a Pois-
an intersection with an average of six cars per minute. To set the son distribution with a mean of 0.01 error per page. What is the
timing of a trafic signal, the following probabilities are used. probability that there are three or fewer errors in 100 pages?
(a) What is the probability that no cars pass through the inter- 3-193. The probability that an individual recovers from an
section within 30 seconds? illness in a one-week time period without treatment is 0.1.
(b) What is the probability that three or more cars pass through Suppose that 20 independent individuals suffering from this
the intersection within 30 seconds? illness are treated with a drug and 4 recover in a one-week time
(c) Calculate the minimum number of cars through the inter- period. If the drug has no effect, what is the probability that
section so that the probability of this number or fewer cars 4 or more people recover in a one-week time period?
in 30 seconds is at least 90%. 3-194. Patient response to a generic drug to control pain is
(d) If the variance of the number of cars through the intersec- scored on a 5-point scale where a 5 indicates complete relief.
tion per minute is 20, is the Poisson distribution appropri- Historically, the distribution of scores is
ate? Explain.
1 2 3 4 5
3-186. A shipment of chemicals arrives in 15 totes. Three 0.05 0.1 0.2 0.25 0.4
of the totes are selected at random without replacement for an