STA 4821 Stochastic Models for CS EC 1 1. There are 3 books on C Programing, 2 books on Python Programing and 1 book on Fortran Programing. In how many ways can these books be arranged on a shelf if {a} the books can be arranged in any order? (bl the Python books must be together and the C books must be together? [c] The C books must be together but the Python and Fortran books can be arranged in any order? 2. There are 3 women and 3 men in movie theater. {a} In how many ways can they sit in a row? (bl In how many ways can they sit in a row ifthe men and women are each to sit together? [c] In how many ways can only the men sit together? (d) In how many ways can they sit together if no two people of the same sex are allowed to sit together? 3. There are 12 students in a graduate class. The students are to be divided into three groups of 3, 4, and 5 members for a class project. How divisions are possible? 4. A committee of 6 is to be chosen from a group consisting of 1' men and 8 women. How many committees are possible? 5. If 3 teachers are to be divided among 4 schools. {a} How many divisions are possible? [b] What if each school must receive two teachers? 6. Suppose thatAand B are mutually exclusive events forwhich P01) = 0.3 and PG?) = 0.5. what is the probability that {a} eitherA or B occurs? (bl A occurs but B does not? (c1 bothAand B occur? 7. In an experiment, A, B, C, and D are events with probabilities P(A U B) = 5/8, P(A) = 3/8, P(C n D) = 1/3 and P(C) = 1/2. Assume that events A and B are disjoint while C and D are independent. Find the following probabilities: (a) P(An B) (b) P(B) (c) P(An B) (d) P(CUD) (e) Are A and B independent? 8. A pair of fair dice is rolled. (a) What is the probability that both dice show the same number? (b) What is the probability that both dice show different numbers? (c) What is the probability that the second die lands on a lower value than does the first? 9. An instructor gives her class the choice to do 7 questions out of the 10 on an exam. (a) How many choices does each student have? (b) How many choices does a student have if he/she must answer at least 3 of the first 5 questions? 10. There are two roads from points A to B and two roads from B to C. Each of the four roads has a 50% chance of being blocked due to an accident, independently of the others. (a) What is the probability that there is an open road from A to C? (b) Now suppose that there is also a direct road from A to C which is independently blocked with the same probability as the roads from A to B and B to C, what is the probability that there is an open road from A to C