91Ó°ÊÓ

A box contains 3 marbles: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box and then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without replacing the first marble.

In an experiment, die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of this experiment? Let \(E_{n}\) denote the event that \(n\) rolls are necessary to complete the experiment. What points of the sample space are contained in \(E_{n} ?\) What is \(\left(\bigcup_{1}^{\infty} E_{n}\right)^{c} ?\)

Two dice are thrown. Let \(E\) be the event that the sum of the dice is odd, let \(F\) be the event that at least one of the dice lands on \(1,\) and let \(G\) be the event that the sum is 5. Describe the events \(E F, E \cup F, F G, E F^{c},\) and \(E F G\).

\(A, B,\) and \(C\) take turns flipping a coin. The first one to get a head wins. The sample space of this experiment can be defined by $$S=\left\\{\begin{array}{l} 1,01,001,0001, \ldots, \\ 0000 \cdots \end{array}\right.$$ (a) Interpret the sample space. (b) Define the following events in terms of \(S:\) (i) \(A\) wins \(=A\) (ii) \(B\) wins \(=B\) (iii) \((A \cup B)^{c}\) Assume that \(A\) flips first, then \(B,\) then \(C,\) then \(A\) and so on.

A system is composed of 5 components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector \(\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right),\) where \(x_{i}\) is equal to 1 if component \(i\) is working and is equal to 0 if component \(i\) is failed. (a) How many outcomes are in the sample space of this experiment? (b) Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both working, or if components \(1,3,\) and 5 are all working. Let \(W\) be the event that the system will work. Specify all the outcomes in \(W\). (c) Let \(A\) be the event that components 4 and 5 are both failed. How many outcomes are contained in the event \(A ?\) (d) Write out all the outcomes in the event \(A W\).

A hospital administrator codes incoming patients suffering gunshot wounds according to whether they have insurance (coding 1 if they do and 0 if they do not) and according to their condition, which is rated as good (g), fair (f), or serious (s). Consider an experiment that consists of the coding of such a patient. (a) Give the sample space of this experiment. (b) Let \(A\) be the event that the patient is in serious condition. Specify the outcomes in \(A\) (c) Let \(B\) be the event that the patient is uninsured. Specify the outcomes in \(B\). (d) Give all the outcomes in the event \(B^{c} \cup A\).

Consider an experiment that consists of determining the type of job-either blue collar or white collarand the political affiliation - Republican, Democratic, or Independent-of the 15 members of an adult soccer team. How many outcomes are (a) in the sample space? (b) in the event that at least one of the team members is a blue-collar worker? (c) in the event that none of the team members considers himself or herself an Independent?

A retail establishment accepts either the American Express or the VISA credit card. A total of 24 percent of its customers carry an American Express card, 61 percent carry a VISA card, and 11 percent carry both cards. What percentage of its customers carry a credit card that the establishment will accept?

Sixty percent of the students at a certain school wear neither a ring nor a necklace. Twenty percent wear a ring and 30 percent wear a necklace. If one of the students is chosen randomly, what is the probability that this student is wearing (a) a ring or a necklace? (b) a ring and a necklace?

A certain town with a population of 100,000 has 3 newspapers: I, II, and III. The proportions of townspeople who read these papers are as follows: I: 10 percent I and II: 8 percent I and II and III: 1 percent II: 30 percent I and III: 2 percent III: 5 percent II and III: 4 percent (The list tells us, for instance, that 8000 people read newspapers I and II.) (a) Find the number of people who read only one newspaper. (b) How many people read at least two newspapers? (c) If I and III are morning papers and II is an evening paper, how many people read at least one morning paper plus an evening paper? (d) How many people do not read any newspapers? (e) How many people read only one morning paper and one evening paper?