91Ó°ÊÓ

Suppose that three events \(-A, B\), and \(C\)-are defined on a sample space \(S\). Use the union, intersection, and complement operations to represent each of the following events: (a) none of the three events occurs (b) all three of the events occur (c) only event \(A\) occurs (d) exactly one event occurs (e) exactly two events occur

What must be true of events \(A\) and \(B\) if (a) \(A \cup B=B\) (b) \(A \cap B=A\)

During orientation week, the latest Spiderman movie was shown twice at State University. Among the entering class of 6000 freshmen, 850 went to see it the first time, 690 the second time, while 4700 failed to see it either time. How many saw it twice?

Let \(A, B\), and \(C\) be any three events. Use Venn diagrams to show that (a) \(A \cup(B \cup C)=(A \cup B) \cup C\) (b) \(A \cap(B \cap C)=(A \cap B) \cap C\)

Use Venn diagrams to suggest an equivalent way of representing the following events: (a) \(\left(A \cap B^{C}\right)^{C}\) (b) \(B \cup(A \cup B)^{C}\) (c) \(A \cap(A \cap B)^{C}\)

A total of twelve hundred graduates of State Tech have gotten into medical school in the past several years. Of that number, one thousand earned scores of twenty-seven or higher on the MCAT and four hundred had GPAs that were \(3.5\) or higher. Moreover, three hundred had MCATs that were twenty-seven or higher and GPAs that were \(3.5\) or higher. What proportion of those twelve hundred graduates got into medical school with an MCAT lower than twenty-seven and a GPA below \(3.5\) ?

Let \(A, B\), and \(C\) be any three events defined on a sample space \(S\). Let \(N(A), N(B), N(C), N(A \cap B)\), \(N(A \cap C), N(B \cap C)\), and \(N(A \cap B \cap C)\) denote the numbers of outcomes in all the different intersections in which \(A, B\), and \(C\) are involved. Use a Venn diagram to suggest a formula for \(N(A \cup B \cup C)\). [Hint: Start with the sum \(N(A)+N(B)+N(C)\) and use the Venn diagram to identify the "adjustments" that need to be made to that sum before it can equal \(N(A \cup B \cup C)\).] As a precedent, note that \(N(A \cup B)=N(A)+N(B)-N(A \cap B)\). There, in the case of two events, subtracting \(N(A \cap B)\) is the "adjustment."

Let \(A\) and \(B\) be any two events defined on \(S\). Suppose that \(P(A)=0.4, P(B)=0.5\), and \(P(A \cap B)=0.1\). What is the probability that \(A\) or \(B\) but not both occur?

Suppose that three fair dice are tossed. Let \(A_{i}\) be the event that a 6 shows on the \(i\) th die, \(i=1,2,3\). Does \(P\left(A_{1} \cup A_{2} \cup A_{3}\right)=\frac{1}{2}\) ? Explain.

Three events \(-A, B\), and \(C\)-are defined on a sample space, \(S\). Given that \(P(A)=0.2, P(B)=0.1\), and \(P(C)=0.3\), what is the smallest possible value for \(P[(A \cup\) \(\left.B \cup C)^{C}\right] ?\)