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The probability that a randomly selected box of a certain type of cereal has a particular prize is \({\rm{.2}}\) . Suppose you purchase box after box until you have obtained two of these prizes.

a. What is the probability that you purchase \({\rm{x}}\) boxes that do not have the desired prize?

b. What is the probability that you purchase four boxes?

c. What is the probability that you purchase at most four boxes?

d. How many boxes without the desired prize do you expect to purchase? How many boxes do you expect to purchase?

A family decides to have children until it has three children of the same gender. Assuming P(B) = P(G) =\({\rm{.5}}\), what is the pmf of X = the number of children in the family?

Three brothers and their wives decide to have children until each family has two female children. What is the pmf of X = the total number of male children born to the brothers? What is E(X), and how does it compare to the expected number of male children born to each brother?

Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses. Suppose this number X has a Poisson distribution with parameter\({\rm{\mu = }}{\rm{.2}}\). (Suggested in 鈥淎verage Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution,鈥� J. Quality Technology,\({\rm{1983 = 126 - 129}}\).) a. What is the probability that a disk has exactly one missing pulse? b. What is the probability that a disk has at least two missing pulses? c. If twodisks are independently selected, what is the probability that neither contains a missing pulse?

An article in the Los Angeles Times (Dec.\({\rm{3, 1993}}\)) reports that\({\rm{1}}\)in\({\rm{200}}\)people carry the defective gene that causes inherited colon cancer. In a sample of\({\rm{1000}}\)individuals, what is the approximate distribution of the number who carry this gene? Use this distribution to calculate the approximate probability that a. Between\({\rm{5}}\)and\({\rm{8}}\)(inclusive) carry the gene. b. At least\({\rm{8}}\)carry the gene.

The Center for Disease Control and Prevention reported in\(2012\)that\(1\)in\(88\)American children had been diagnosed with an autism spectrum disorder (ASD).

a. If a random sample of\(200\)American children is selected, what are the expected value and standard deviation of the number who have been diagnosed with ASD?

b. Referring back to (a), calculate the approximate probability that at least\(2\)children in the sample have been diagnosed with ASD?

c. If the sample size is\(352\), what is the approximate probability that fewer than\(5\)of the selected children have been diagnosed with ASD?

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate\(\alpha = 8\)per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter\(\mu = 8t\). a. What is the probability that exactly 6 small aircraft arrive during a one-hour period? At least\(6\)? At least\({\bf{10}}\)? b. What are the expected value and standard deviations of the number of small aircraft that arrive during a\({\bf{90}}\)-min period? c. What is the probability that at least\({\bf{20}}\)small aircraft arrive during a\({\bf{2}}.{\bf{5}}\)-hour period? That at most\({\bf{10}}\)arrive during this period?

Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of\({\bf{10}}\)organisms/m3 (the article 鈥淐ounting at Low Concentrations: The Statistical Challenges of Verifying Ballast Water Discharge Standards鈥� (Ecological Applications) considers using the Poisson process for this purpose). a. What is the probability that one cubic meter of discharge contains at least 8 organisms? b. What is the probability that the number of organisms in\({\bf{1}}.{\bf{5}}{\rm{ }}{{\bf{m}}^3}\)of discharge exceeds its mean value by more than one standard deviation? c. For what amount of discharge would the probability of containing at least one organism be\(.{\bf{999}}\)?

The number of requests for assistance received by a towing service is a Poisson process with rate \({\rm{\alpha = 4}}\) per hour. a. Compute the probability that exactly ten requests are received during a particular \({\rm{2}}\)-hour period. b. If the operators of the towing service take a \({\rm{30}}\)-min break for lunch, what is the probability that they do not miss any calls for assistance? c. How many calls would you expect during their break?

In proof testing of circuit boards, the probability that any particular diode will fail is\(.{\bf{01}}\). Suppose a circuit board contains\({\bf{200}}\)diodes. a. How many diodes would you expect to fail, and what is the standard deviation of the number that is expected to fail? b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board? c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (Aboard works properly only if all its diodes work.)