Chapter 1

Let \(\Omega\) consist of the set of positive integers. Consider the subsets $$A=\\{\omega: \omega \leq 12\\} \quad B=\\{\omega: \omega<8\\} \quad C=\\{\omega: \omega \text { is even }\\}$$ \(D=\\{\omega: \omega\) is a multiple of 3\(\\} \quad E=\\{\omega: \omega\) is a multiple of 4\(\\}\) Describe in terms of \(A, B, C, D, E\) and their complements the following sets: a. \\{1,3,5,7\\} b. \\{3,6,9\\} c. \\{8,10\\} d. The even integers greater than 12 . e. The positive integers which are multiples of six. f. The integers which are even and no greater than 6 or which are odd and greater than 12 .

A group of five persons consists of two men and three women. They are selected one-by-one in a random manner. Let \(E_{i}\) be the event a man is selected on the ith selection. Write an expression for the event that both men have been selected by the third selection.

Find the (classical) probability that among three random digits, with each digit (0 through 9 ) being equally likely and each triple equally likely: a. All three are alike. b. No two are alike. c. The first digit is 0 . d. Exactly two are alike.

A committee of five is chosen from a group of 20 people. What is the probability that a specified member of the group will be on the committee?

Ten employees of a company drive their cars to the city each day and park randomly in ten spots. What is the (classical) probability that on a given day Jim will be in place three? There are \(n !\) equally likely ways to arrange \(n\) items (order important).

John thinks the probability the Houston Texans will win next Sunday is 0.3 and the probability the Dallas Cowboys will win is 0.7 (they are not playing each other). He thinks the probability both will win is somewhere between-say, \(0.5 .\) Is that a reasonable assumption? Justify your answer.

Suppose \(P(A)=0.5\) and \(P(B)=0.3 .\) What is the largest possible value of \(P(A B) ?\) Using the maximum value of \(P(A B),\) determine \(P\left(A B^{c}\right), P\left(A^{c} B\right), P\left(A^{c} B^{c}\right)\) and \(P(A \cup B)\). Are these values determined uniquely?

The class \(\\{A, B, C\\}\) of events is a partition. Event \(A\) is twice as likely as \(C\) and event \(B\) is as likely as the combination \(A\) or \(C\). Determine the probabilities \(P(A), P(B), P(C)\).

Determine the probability \(P(A \cup B \cup C)\) in terms of the probabilities of the events \(A, B, C\) and their intersections.

The set combination \(A \oplus B=A B^{c} \bigvee A^{c} B\) is known as the disjunctive union or the symetric difference of \(A\) and \(B\). This is the event that only one of the events \(A\) or \(B\) occurs on a trial. Determine \(P(A \oplus B)\) in terms of \(P(A), P(B)\), and \(P(A B)\).

Use fundamental properties of probability to show a. \(P(A B) \leq P(A) \leq P(A \cup B) \leq P(A)+P(B)\) b. \(P\left(\bigcap_{j=1}^{\infty} E_{j}\right) \leq P\left(E_{i}\right) \leq P\left(\bigcup_{j=1}^{\infty} E_{j}\right) \leq \sum_{j=1}^{\infty} P\left(E_{j}\right)\)

Suppose \(P_{1}, P_{2}\) are probability measures and \(c_{1}, c_{2}\) are positive numbers such that \(c_{1}+c_{2}=1\). Show that the assignment \(P(E)=c_{1} P_{1}(E)+c_{2} P_{2}(E)\) to the class of events is a probability measure. Such a combination of probability measures is known as a mixture. Extend this to \(P(E)=\sum_{i=1}^{n} c_{i} P_{i}(E),\) where the \(P_{i}\) are probabilities measures, \(c_{i}>0,\) and \(\sum_{i=1}^{n} c_{i}=1\)