Probability Theory and Random Process1. A fair die is tossed n times and each time the number appearing on the upward face is recorded.

What is the probability that the sum of numbers appearing on upward faces is n + 2? (The answer,of course, is not a number but rather an expression involving n).2. Alice has n keys, of which one will open her door.

(a) If she tries the keys at random, discarding those that do not work, show that the probability

that she will open the door on her kth attempt (where 1 ≤ k ≤ n) is 1/n.

(b) You should not be surprised that the above probably does not depend on k. If you are, then

maybe you were thinking of a different question, namely “What is the probability that Alice

will find the key in her first k attempts?”. Show that this probability is k/n.

(c) Suppose we change the settings, and Alice does not discard previously tried keys – obviously, a

bad strategy. Can we still answer the questions in parts (a) and (b)? Yes (this will be in a future

homework assignment)! But we can no longer deploy counting methods. The reason is simple:

now the sample space is not finite. Describe the sample space for this new experiment.

3. Sometimes order matters, but sometimes it does not! An urn contains n balls, of which

one is special. Of these n balls, k is to be randomly selected, where 1 < k < n. We consider two

different ways of randomly choosing balls from the urn:

(a) k balls are simultaneously withdrawn, with each ball in the urn being equally likely to be picked;

(b) the k balls are withdrawn one at a time, with each selection being equally likely to be any of

the balls that remain at the time.

For each of the two sampling methods, show that the probability that the special ball is selected is

the same, and equal to k

n.

To answer part (a) you have to define the sample space in a such way that the order in which the

k balls are selected does not matter. In part (b) you instead need to account for the order in which

the balls are withdrawn.

4. A pair of dice is rolled until a sum of 5 or 7 appears. Find the probability that a 5 occurs first. Hint:

Let A denote the event that a 5 occurs first and let An be the event that a 5 occurs on the nth roll

and no 5 or 7 occurs on the first n − 1 rolls. Then A = ∪nAn. To compute each P(An) you may

assume that, for every n > 1, each of the 36n possible outcomes of the experiment of rolling the pair

of dice n times are equally likely.

5. Your statistics teacher announces a thirty-page assignment on Monday that is to be finished by

Friday. You intend to read the first n1 pages on Monday, the next n2 pages on Tuesday, the next n3

pages on Wednesday, and the final n4 pages on Thursday, where n1 + n2 + n3 + n4 = 30 and each

ni ≥ 1. In how many ways can you complete the assignment?

1

6. Suppose that n people are to be randomly arranged. Two of them, Alice and Bob are friends and

hope to be seated next to each other.

(a) What is the probability that Alice and Bob end up sitting next to each other if the n people

are arranged in a line?

(b) What would the probability be if the people are randomly arranged in a circle?

7. A bin of 50 manufactured parts contains three defective parts and 47 non-defective parts. A sample of

six parts is selected from the 50 parts. Write down an expression for the number of different samples

of size six that contain exactly two defective parts.

8. An elevator in a building starts with five passengers and stops at seven floors. If every passenger is

equally likely to get off at each floor and all the passengers leave independently of each other, what

is the probability that no two passengers will get off at the same floor? For more information on Probability Theory check on this: https://en.wikipedia.org/wiki/Probability_theory

Try it now!

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

Australia Assessments has gained an international reputation of being the leading website in custom assignment writing services. Once you give us the instructions of your paper through the order form, we will complete the rest.

Essays

As we work towards providing the best custom assignment services, our company provides assignment services for any type of academic essay. We will help you develop professionally written essays that are rich in content and free from plagiarism.

Admissions

Admission and Business Papers

Our skilled team of professional writers will ensure that we help you craft a remarkable admission essay for your desired Master's program in your institution of choice. We won't stop there. Once you enter the job market, we will be available to secure you a position at your desired worksite by creating an outstanding portfolio or resume.

Editing

Editing and Proofreading

Our editorial team is always available for all editing and proofreading services. They check completed papers by our writers and also provide professional opinions to papers completed by our clients.

Coursework

Technical papers

We harbor professional academic writers with different qualifications in diverse academic fields. As such, we are capable of handling both simple and technical papers. Ensure that you provide us with correct and complete instructions in the order form.