Let [math]\Omega = \{a,b,c\}[/math] be a sample space. Let [math]m(a) = 1/2[/math], [math]m(b) = 1/3[/math], and [math]m(c) = 1/6[/math]. Find the probabilities for all eight subsets of [math]\Omega[/math].

Give a possible sample space [math]\Omega[/math] for each of the following experiments:

• An election decides between two candidates A and B.
• A two-sided coin is tossed.
• A student is asked for the month of the year and the day of the week on which her birthday falls.
• A student is chosen at random from a class of ten students.
• You receive a grade in this course.

For which of the cases in Exercise would it be reasonable to assign the uniform distribution function?

Describe in words the events specified by the following subsets of

(see Example).

• [math]E = \{\mbox{HHH,HHT,HTH,HTT}\}[/math].
• [math]E = \{\mbox{HHH,TTT}\}[/math].
• [math]E = \{\mbox{HHT,HTH,THH}\}[/math].
• [math]E = \{\mbox{HHT,HTH,HTT,THH,THT,TTH,TTT}\}[/math].

What are the probabilities of the events described in Exercise?

A die is loaded in such a way that the probability of each face turning up is proportional to the number of dots on that face. (For example, a six is three times as probable as a two.) What is the probability of getting an even number in one throw?

Let [math]A[/math] and [math]B[/math] be events such that [math]P(A \cap B) = 1/4[/math], [math]P(\tilde A) = 1/3[/math], and [math]P(B) = 1/2[/math]. What is [math]P(A \cup B)[/math]?

A student must choose one of the subjects, art, geology, or psychology, as an elective. She is equally likely to choose art or psychology and twice as likely to choose geology. What are the respective probabilities that she chooses art, geology, and psychology?

A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability 5/8, French with probability 5/8, and art and French together with probability 1/4. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French?

For a bill to come before the president of the United States, it must be passed by both the House of Representatives and the Senate. Assume that, of the bills presented to these two bodies, 60 percent pass the House, 80 percent pass the Senate, and 90 percent pass at least one of the two. Calculate the probability that the next bill presented to the two groups will come before the president.