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Introductory Probability and StatisticsHome assignment 2 14

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Introductory Probability and StatisticsHome assignment 2 14.09.2020 Submit solutions to your class teachers before the class on the week beginning 21.09.2020.

Problem 1. Websites often set requirements on users' passwords strength. For example, a password must be at least 8 characters long, or to have at least one capital letter, etc.

Suppose a user wants to set a password consisting of n characters, each can be either a small or capital Latin letter, or a digit; no other symbols are allowed. Find the number of different passwords of length n if

(a) a password may consist of any digits and letters in any order;

(b) a password must have at least one digit and at least one small or capital letter.

(c) a password must consist of only digits in the non-decreasing order.

Problem 2. A retail company has hired 10 new employees and wants to send them to work in its stores. In how many ways can this be done

1. (a) if the company has 2 stores?
2. (b) if the company has 2 stores and wants to send 5 new employees to each of them?
3. (c) if the company has one main store, where it wants to send 6 new employees, and 2 local stores, to each of which it wants to send 2 new employees?

Problem 3. (Birthday problem) Find the probability that in a group of n people at least two have their birthdays on the same day (the year of birth may be different). Assume that there are 365 days in the year and each is equally probable for a birthday.

What is the minimal n such that this probability is 50%? 99%? (This problem is often called Birthday paradox because n turns out to be quite small.)

Problem 4. An economics student must choose 4 courses to attend in the forthcoming academic year. There are 8 courses offered by different lectures, as shown in the table below.

Lecturer

Prof. V. Lowmark(Microeconomics, macroeconomics, nGame theory)

Prof. A. Greedman(Banking Stock markets)

Dr. T. Kantan (Asian economics)

Dr. I. Workmore (Developing countries Labour economics)

 

Stock markets

Asian economics Developing countries Labour economics

1.    (a) In how many ways can he choose the courses?
2. (b) Prof. Lowmark is the head of the Economics Department, so the student thinks it may be a
3. good idea to take his course. However, his exams are very difficult, so the student wants to
4. take just one of his courses. Given this, in how many ways can he arrange his curriculum?
5. (c) Tired of thinking, the student decides to choose the courses just randomly. What is the
6. probability to choose at least one course of Prof. Lowmark? At least two of his courses?

Problem 5. Consider the following game. There are n people standing in a queue, and there are m ≥ n closed boxes, k of the boxes contain prizes. Each next person in the queue can open any box and if it contains a prize this person will receive the prize. One person can open only one box. What is the best position in the queue to maximize the probability of winning a prize? Hint: try to compute this probability for the first and the second positions in the queue.

Additional problems

Durrett: p. 71, problems 2, 12, 17, 24∗, 32, 60, 76